Processing math: 24%

Your data matches 528 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000319: Integer partitions ⟶ ℤResult quality: 75% values known / values provided: 99%distinct values known / distinct values provided: 75%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 4 = 3 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> ? = 1 + 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 4 + 1
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> ? = 4 + 1
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 6 + 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? = 9 + 1
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [6,4,2]
=> [3,3,2,2,1,1]
=> ? = 2 + 1
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [5,4,2]
=> [3,3,2,2,1]
=> ? = 2 + 1
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3 + 1
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3 + 1
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3 + 1
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> ? = 0 + 1
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [7,5,4,1]
=> [4,3,3,3,2,1,1]
=> ? = 5 + 1
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [6,5,3,1]
=> [4,3,3,2,2,1]
=> ? = 4 + 1
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [7,5,4,1]
=> [4,3,3,3,2,1,1]
=> ? = 5 + 1
([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> [8,6,4,2]
=> [4,4,3,3,2,2,1,1]
=> ? = 6 + 1
([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> ? = 1 + 1
([(0,9),(1,10),(1,18),(2,10),(2,17),(3,17),(3,18),(5,14),(6,15),(7,12),(7,13),(8,7),(9,1),(9,2),(9,3),(10,8),(11,14),(11,15),(12,19),(13,19),(14,12),(14,16),(15,13),(15,16),(16,19),(17,5),(17,11),(18,6),(18,11),(19,4)],20)
=> [9,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> ? = 1 + 1
([(0,13),(0,14),(1,16),(2,15),(3,17),(4,19),(5,18),(6,12),(6,16),(7,6),(8,2),(8,22),(9,1),(9,22),(10,4),(11,3),(11,23),(12,5),(12,20),(13,7),(14,8),(14,9),(15,23),(16,20),(17,21),(18,21),(20,17),(20,18),(21,19),(22,11),(22,15),(23,10)],24)
=> [9,7,5,3]
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> [5,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 5 + 1
([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> [6,4,4,3,2,1]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> ? = 6 + 1
([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> [6,4,3,3,1,1]
=> [4,3,3,1,1]
=> [5,3,3,1]
=> ? = 5 + 1
([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> [6,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 5 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> [6,4,4,3,2,1]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> [6,4,3,3,1,1]
=> [4,3,3,1,1]
=> [5,3,3,1]
=> ? = 5 + 1
([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> [6,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 5 + 1
([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> [6,4,4,2,2]
=> [4,4,2,2]
=> [4,4,2,2]
=> ? = 4 + 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition λ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of λ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let λ=(5,5,4,4,2,1). Removing the border strips successively yields the sequence of partitions (5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(). The first strip (5,5,4,4,2,1)(4,3,3,1) crosses 4 times, the second strip (4,3,3,1)(2,2) crosses 3 times, the strip (2,2)(1) crosses 1 time, and the remaining strip (1)() does not cross. This yields the spin of (5,5,4,4,2,1) to be 4+3+1=8.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000320: Integer partitions ⟶ ℤResult quality: 75% values known / values provided: 99%distinct values known / distinct values provided: 75%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 1 = 0 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 4 = 3 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> ? = 1 + 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 4 + 1
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,3,1]
=> [4,3,3,2,1]
=> ? = 4 + 1
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> ? = 6 + 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? = 9 + 1
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [6,4,2]
=> [3,3,2,2,1,1]
=> ? = 2 + 1
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [5,4,2]
=> [3,3,2,2,1]
=> ? = 2 + 1
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3 + 1
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3 + 1
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3 + 1
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> ? = 0 + 1
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [7,5,4,1]
=> [4,3,3,3,2,1,1]
=> ? = 5 + 1
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [6,5,3,1]
=> [4,3,3,2,2,1]
=> ? = 4 + 1
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [7,5,4,1]
=> [4,3,3,3,2,1,1]
=> ? = 5 + 1
([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> [8,6,4,2]
=> [4,4,3,3,2,2,1,1]
=> ? = 6 + 1
([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> ? = 1 + 1
([(0,9),(1,10),(1,18),(2,10),(2,17),(3,17),(3,18),(5,14),(6,15),(7,12),(7,13),(8,7),(9,1),(9,2),(9,3),(10,8),(11,14),(11,15),(12,19),(13,19),(14,12),(14,16),(15,13),(15,16),(16,19),(17,5),(17,11),(18,6),(18,11),(19,4)],20)
=> [9,5,5,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> ? = 1 + 1
([(0,13),(0,14),(1,16),(2,15),(3,17),(4,19),(5,18),(6,12),(6,16),(7,6),(8,2),(8,22),(9,1),(9,22),(10,4),(11,3),(11,23),(12,5),(12,20),(13,7),(14,8),(14,9),(15,23),(16,20),(17,21),(18,21),(20,17),(20,18),(21,19),(22,11),(22,15),(23,10)],24)
=> [9,7,5,3]
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> [5,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 5 + 1
([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> [6,4,4,3,2,1]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> ? = 6 + 1
([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> [6,4,3,3,1,1]
=> [4,3,3,1,1]
=> [5,3,3,1]
=> ? = 5 + 1
([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> [6,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 5 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> [6,4,4,3,2,1]
=> [4,4,3,2,1]
=> [5,4,3,2]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> [6,4,3,3,1,1]
=> [4,3,3,1,1]
=> [5,3,3,1]
=> ? = 5 + 1
([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> [6,3,3,3,1,1]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 5 + 1
([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> [6,4,4,2,2]
=> [4,4,2,2]
=> [4,4,2,2]
=> ? = 4 + 1
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition λ=(λ1,,λk) can be decomposed into border strips. For 0j<λ1 let nj be the length of the border strip starting at (λ1j,0). The dinv adjustment is then defined by j:nj>0(λ11j). The following example is taken from Appendix B in [2]: Let λ=(5,5,4,4,2,1). Removing the border strips successively yields the sequence of partitions (5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(), and we obtain (n0,,n4)=(10,7,0,3,1). The dinv adjustment is thus 4+3+1+0=8.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000676: Dyck paths ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 0 + 3
([],4)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([],6)
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 3
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [4,3,3,3,3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 + 3
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 6 + 3
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [6,5,5,4,4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 9 + 3
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 3
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 + 3
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 3
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [3,3,3,3,2,2,1,1,1]
=> ?
=> ? = 0 + 3
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [3,3,3,3,2,2,1,1,1,1]
=> ?
=> ? = 0 + 3
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [4,4,3,3,2,2,1,1]
=> ?
=> ? = 2 + 3
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [3,3,3,2,2,2,1,1,1]
=> ?
=> ? = 0 + 3
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [3,3,3,2,2,2,2,1,1,1]
=> ?
=> ? = 0 + 3
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [3,3,3,2,2,2,2,1,1,1,1]
=> ?
=> ? = 0 + 3
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [4,4,3,3,2,1,1,1]
=> ?
=> ? = 2 + 3
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [4,4,4,3,3,2,1,1,1]
=> ?
=> ? = 3 + 3
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [4,4,4,3,3,2,2,1,1,1]
=> ?
=> ? = 3 + 3
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [4,4,4,3,2,2,1,1,1]
=> ?
=> ? = 3 + 3
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [3,3,3,3,2,2,1,1]
=> ?
=> ? = 0 + 3
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [3,3,3,3,2,2,2,1,1]
=> ?
=> ? = 0 + 3
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [5,4,4,4,3,2,2,1,1]
=> ?
=> ? = 5 + 3
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [5,4,4,3,3,2,1,1]
=> ?
=> ? = 4 + 3
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [5,4,4,4,3,2,2,1,1]
=> ?
=> ? = 5 + 3
([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> [5,5,4,4,3,3,2,2,1,1]
=> ?
=> ? = 6 + 3
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> [4,3,3,3,3,1,1,1]
=> ?
=> ? = 1 + 3
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength n with k up steps in odd positions and k returns to the main diagonal are counted by the binomial coefficient \binom{n-1}{k-1} [3,4].
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001039: Dyck paths ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 0 + 3
([],4)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 1 + 3
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 0 + 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 2 + 3
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 1 + 3
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 0 + 3
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 0 + 3
([],6)
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 3 + 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 2 + 3
([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 3
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 3
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [4,3,3,3,3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 3
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 3 + 3
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 6 + 3
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [6,5,5,4,4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 9 + 3
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 3
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 3
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5 + 3
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 3
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [3,3,3,3,2,2,1,1,1]
=> ?
=> ? = 0 + 3
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [3,3,3,3,2,2,1,1,1,1]
=> ?
=> ? = 0 + 3
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [4,4,3,3,2,2,1,1]
=> ?
=> ? = 2 + 3
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [3,3,3,2,2,2,1,1,1]
=> ?
=> ? = 0 + 3
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [3,3,3,2,2,2,2,1,1,1]
=> ?
=> ? = 0 + 3
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [3,3,3,2,2,2,2,1,1,1,1]
=> ?
=> ? = 0 + 3
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [4,4,3,3,2,1,1,1]
=> ?
=> ? = 2 + 3
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [4,4,4,3,3,2,1,1,1]
=> ?
=> ? = 3 + 3
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [4,4,4,3,3,2,2,1,1,1]
=> ?
=> ? = 3 + 3
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [4,4,4,3,2,2,1,1,1]
=> ?
=> ? = 3 + 3
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [3,3,3,3,2,2,1,1]
=> ?
=> ? = 0 + 3
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [3,3,3,3,2,2,2,1,1]
=> ?
=> ? = 0 + 3
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [5,4,4,4,3,2,2,1,1]
=> ?
=> ? = 5 + 3
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [5,4,4,3,3,2,1,1]
=> ?
=> ? = 4 + 3
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [5,4,4,4,3,2,2,1,1]
=> ?
=> ? = 5 + 3
([(0,2),(0,3),(1,9),(1,12),(2,1),(3,5),(3,8),(4,28),(5,24),(6,22),(7,17),(8,13),(8,24),(9,23),(10,15),(10,16),(11,20),(11,21),(12,23),(12,27),(13,11),(13,26),(13,27),(14,29),(15,29),(16,7),(16,29),(18,16),(19,14),(20,19),(21,18),(22,10),(22,18),(23,4),(23,25),(24,6),(24,26),(25,19),(25,28),(26,21),(26,22),(27,20),(27,25),(28,14),(28,15),(29,17)],30)
=> [10,8,6,4,2]
=> [5,5,4,4,3,3,2,2,1,1]
=> ?
=> ? = 6 + 3
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 3
([(0,2),(1,8),(2,5),(2,6),(2,7),(3,17),(4,16),(5,12),(5,13),(6,12),(6,14),(7,13),(7,14),(8,10),(8,11),(9,18),(10,18),(11,18),(12,1),(13,4),(13,15),(14,3),(14,15),(15,16),(15,17),(16,9),(16,10),(17,9),(17,11)],19)
=> [8,5,5,1]
=> [4,3,3,3,3,1,1,1]
=> ?
=> ? = 1 + 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000157
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00106: Standard tableaux catabolismStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 1 = 0 + 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1 = 0 + 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> 3 = 2 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1 = 0 + 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2],[3],[4],[5],[6]]
=> 4 = 3 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ?
=> ? = 1 + 1
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 1 + 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21,22,23],[24]]
=> ?
=> ? = 1 + 1
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 3 + 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24],[25]]
=> ?
=> ? = 4 + 1
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15,16],[17,18,19],[20]]
=> ?
=> ? = 4 + 1
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ?
=> ? = 6 + 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18,19,20],[21,22,23,24,25,26,27],[28,29,30,31,32],[33,34,35],[36]]
=> ?
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ?
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ?
=> ? = 4 + 1
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ?
=> ? = 5 + 1
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? = 0 + 1
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ?
=> ? = 0 + 1
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ?
=> ? = 1 + 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ?
=> ? = 0 + 1
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ?
=> ? = 0 + 1
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ?
=> ? = 0 + 1
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ?
=> ? = 0 + 1
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
=> ?
=> ? = 0 + 1
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19]]
=> ?
=> ? = 0 + 1
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16],[17,18,19,20]]
=> ?
=> ? = 0 + 1
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 0 + 1
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ?
=> ? = 0 + 1
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18],[19,20]]
=> ?
=> ? = 2 + 1
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ?
=> ? = 0 + 1
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
=> ?
=> ? = 0 + 1
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18]]
=> ?
=> ? = 0 + 1
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20]]
=> ?
=> ? = 0 + 1
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21]]
=> ?
=> ? = 0 + 1
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ?
=> ? = 0 + 1
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16,17],[18,19]]
=> ?
=> ? = 2 + 1
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19,20],[21,22,23]]
=> ?
=> ? = 3 + 1
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20,21,22],[23,24,25]]
=> ?
=> ? = 3 + 1
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19],[20,21,22]]
=> ?
=> ? = 3 + 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ?
=> ? = 0 + 1
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18]]
=> ?
=> ? = 0 + 1
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20]]
=> ?
=> ? = 0 + 1
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ?
=> ? = 5 + 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ?
=> ? = 0 + 1
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ?
=> ? = 0 + 1
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18,19],[20,21,22],[23]]
=> ?
=> ? = 4 + 1
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ?
=> ? = 5 + 1
Description
The number of descents of a standard tableau. Entry i of a standard Young tableau is a descent if i+1 appears in a row below the row of i.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00134: Standard tableaux descent wordBinary words
St000288: Binary words ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 2 = 0 + 2
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 3 = 1 + 2
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2 = 0 + 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2 = 0 + 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2 = 0 + 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2 = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 2 = 0 + 2
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 4 = 2 + 2
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 3 = 1 + 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 2 = 0 + 2
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 2 = 0 + 2
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 11111 => 5 = 3 + 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 4 = 2 + 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 4 = 2 + 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 4 = 2 + 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 4 = 2 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 4 = 2 + 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,3,4,8,9,15,16],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? = 1 + 2
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> 10010010000 => ? = 1 + 2
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [[1,3,4,5,6,12,13,21,22,23,24],[2,8,9,10,11,19,20],[7,15,16,17,18],[14]]
=> ? => ? = 1 + 2
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> 10100100010000 => ? = 3 + 2
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [[1,3,4,8,9,15,16,24,25],[2,6,7,13,14,22,23],[5,11,12,20,21],[10,18,19],[17]]
=> ? => ? = 4 + 2
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [[1,3,4,8,13,19,20],[2,6,7,12,18],[5,10,11,17],[9,15,16],[14]]
=> ? => ? = 4 + 2
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? => ? = 6 + 2
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [[1,3,4,8,9,15,16,24,25,35,36],[2,6,7,13,14,22,23,33,34],[5,11,12,20,21,31,32],[10,18,19,29,30],[17,27,28],[26]]
=> ? => ? = 9 + 2
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> 01010010000 => ? = 2 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,9,13,14],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 4 + 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> 01010010000 => ? = 2 + 2
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> 1010010000 => ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? = 3 + 2
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> 1010010000 => ? = 1 + 2
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> 1010010000 => ? = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,9,13,14],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 4 + 2
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,3,10,14,15],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? = 5 + 2
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> 01000100000 => ? = 0 + 2
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,7,8,14,15],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 0 + 2
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,3,4,8,9,15,16],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? = 1 + 2
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
=> ? => ? = 0 + 2
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,3,4,5,6,12,13,14],[2,8,9,10,11],[7]]
=> ? => ? = 0 + 2
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,3,4,5,6,12,13,14,15],[2,8,9,10,11],[7]]
=> ? => ? = 0 + 2
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,7,12,13,14],[4,5,6,11],[8,9,10]]
=> ? => ? = 0 + 2
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,7,8,14,15,16],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 0 + 2
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [[1,2,3,4,9,10,17,18,19],[5,6,7,8,15,16],[11,12,13,14]]
=> ? => ? = 0 + 2
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [[1,2,3,4,9,10,17,18,19,20],[5,6,7,8,15,16],[11,12,13,14]]
=> ? => ? = 0 + 2
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 0 + 2
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,3,4,5,6,12,13],[2,8,9,10,11],[7]]
=> ? => ? = 0 + 2
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [[1,2,5,6,11,12,19,20],[3,4,9,10,17,18],[7,8,15,16],[13,14]]
=> ? => ? = 2 + 2
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,7,8,14,15],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 0 + 2
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,5,6,7,13,14,15],[3,4,10,11,12],[8,9]]
=> ? => ? = 0 + 2
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [[1,2,3,7,8,9,16,17,18],[4,5,6,13,14,15],[10,11,12]]
=> ? => ? = 0 + 2
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [[1,2,3,7,8,9,10,18,19,20],[4,5,6,14,15,16,17],[11,12,13]]
=> ? => ? = 0 + 2
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [[1,2,3,7,8,9,10,18,19,20,21],[4,5,6,14,15,16,17],[11,12,13]]
=> ? => ? = 0 + 2
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,5,6,11,12,13],[3,4,9,10],[7,8]]
=> ? => ? = 0 + 2
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [[1,2,5,6,11,17,18,19],[3,4,9,10,16],[7,8,14,15],[12,13]]
=> ? => ? = 2 + 2
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [[1,2,3,7,8,14,21,22,23],[4,5,6,12,13,20],[9,10,11,18,19],[15,16,17]]
=> ? => ? = 3 + 2
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [[1,2,3,7,8,14,15,23,24,25],[4,5,6,12,13,21,22],[9,10,11,19,20],[16,17,18]]
=> ? => ? = 3 + 2
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [[1,2,3,7,12,13,20,21,22],[4,5,6,11,18,19],[8,9,10,17],[14,15,16]]
=> ? => ? = 3 + 2
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,5,6,7,13,14],[3,4,10,11,12],[8,9]]
=> ? => ? = 0 + 2
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [[1,2,3,4,9,10,17,18],[5,6,7,8,15,16],[11,12,13,14]]
=> ? => ? = 0 + 2
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [[1,2,3,4,9,10,11,19,20],[5,6,7,8,16,17,18],[12,13,14,15]]
=> ? => ? = 0 + 2
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [[1,3,4,5,10,16,17,25,26],[2,7,8,9,15,23,24],[6,12,13,14,22],[11,19,20,21],[18]]
=> ? => ? = 5 + 2
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,5,6,7,13,14],[3,4,10,11,12],[8,9]]
=> ? => ? = 0 + 2
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,7,8,9,16,17],[4,5,6,13,14,15],[10,11,12]]
=> ? => ? = 0 + 2
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [[1,3,4,8,9,15,22,23],[2,6,7,13,14,21],[5,11,12,19,20],[10,17,18],[16]]
=> ? => ? = 4 + 2
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [[1,3,4,5,10,16,17,25,26],[2,7,8,9,15,23,24],[6,12,13,14,22],[11,19,20,21],[18]]
=> ? => ? = 5 + 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000546
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000546: Permutations ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 0 + 2
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 1 + 2
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 0 + 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 0 + 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 0 + 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 0 + 2
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 2 + 2
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 1 + 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 0 + 2
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 0 + 2
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 3 + 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 2 + 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 2 + 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 2 + 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 2 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 2 + 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? => ? = 1 + 2
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1 + 2
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21,22,23],[24]]
=> ? => ? = 1 + 2
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 3 + 2
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24],[25]]
=> ? => ? = 4 + 2
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15,16],[17,18,19],[20]]
=> ? => ? = 4 + 2
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ? => ? = 6 + 2
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18,19,20],[21,22,23,24,25,26,27],[28,29,30,31,32],[33,34,35],[36]]
=> ? => ? = 9 + 2
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 2 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? => ? = 4 + 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 2 + 2
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3 + 2
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 2
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? => ? = 4 + 2
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ? => ? = 5 + 2
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 0 + 2
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? => ? = 0 + 2
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? => ? = 1 + 2
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? => ? = 0 + 2
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? => ? = 0 + 2
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ? => ? = 0 + 2
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? => ? = 0 + 2
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
=> ? => ? = 0 + 2
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19]]
=> ? => ? = 0 + 2
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16],[17,18,19,20]]
=> ? => ? = 0 + 2
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? = 0 + 2
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ? => ? = 0 + 2
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18],[19,20]]
=> ? => ? = 2 + 2
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? => ? = 0 + 2
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
=> ? => ? = 0 + 2
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18]]
=> ? => ? = 0 + 2
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20]]
=> ? => ? = 0 + 2
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21]]
=> ? => ? = 0 + 2
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ? => ? = 0 + 2
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16,17],[18,19]]
=> ? => ? = 2 + 2
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19,20],[21,22,23]]
=> ? => ? = 3 + 2
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20,21,22],[23,24,25]]
=> ? => ? = 3 + 2
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19],[20,21,22]]
=> ? => ? = 3 + 2
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? => ? = 0 + 2
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18]]
=> ? => ? = 0 + 2
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20]]
=> ? => ? = 0 + 2
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ? => ? = 5 + 2
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? => ? = 0 + 2
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ? => ? = 0 + 2
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18,19],[20,21,22],[23]]
=> ? => ? = 4 + 2
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ? => ? = 5 + 2
Description
The number of global descents of a permutation. The global descents are the integers in the set C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}. In particular, if i\in C(\pi) then i is a descent. For the number of global ascents, see [[St000234]].
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00106: Standard tableaux catabolismStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 2 = 0 + 2
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> 3 = 1 + 2
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 2 = 0 + 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 2 = 0 + 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 2 = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 2 = 0 + 2
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> 4 = 2 + 2
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 3 = 1 + 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 2 = 0 + 2
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2],[3],[4],[5],[6]]
=> 5 = 3 + 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 2 + 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 2 + 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 2 + 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 2 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 2 + 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ?
=> ? = 1 + 2
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12]]
=> ? = 1 + 2
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21,22,23],[24]]
=> ?
=> ? = 1 + 2
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 3 + 2
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24],[25]]
=> ?
=> ? = 4 + 2
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15,16],[17,18,19],[20]]
=> ?
=> ? = 4 + 2
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ?
=> ? = 6 + 2
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18,19,20],[21,22,23,24,25,26,27],[28,29,30,31,32],[33,34,35],[36]]
=> ?
=> ? = 9 + 2
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 2 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ?
=> ? = 4 + 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 2 + 2
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 3 + 2
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 1 + 2
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 1 + 2
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ?
=> ? = 4 + 2
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12],[13]]
=> ? = 3 + 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ?
=> ? = 5 + 2
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? = 0 + 2
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ?
=> ? = 0 + 2
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ?
=> ? = 1 + 2
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ?
=> ? = 0 + 2
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ?
=> ? = 0 + 2
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ?
=> ? = 0 + 2
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ?
=> ? = 0 + 2
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
=> ?
=> ? = 0 + 2
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19]]
=> ?
=> ? = 0 + 2
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16],[17,18,19,20]]
=> ?
=> ? = 0 + 2
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 0 + 2
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ?
=> ? = 0 + 2
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18],[19,20]]
=> ?
=> ? = 2 + 2
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ?
=> ? = 0 + 2
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
=> ?
=> ? = 0 + 2
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18]]
=> ?
=> ? = 0 + 2
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20]]
=> ?
=> ? = 0 + 2
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21]]
=> ?
=> ? = 0 + 2
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ?
=> ? = 0 + 2
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16,17],[18,19]]
=> ?
=> ? = 2 + 2
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19,20],[21,22,23]]
=> ?
=> ? = 3 + 2
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20,21,22],[23,24,25]]
=> ?
=> ? = 3 + 2
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19],[20,21,22]]
=> ?
=> ? = 3 + 2
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ?
=> ? = 0 + 2
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18]]
=> ?
=> ? = 0 + 2
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20]]
=> ?
=> ? = 0 + 2
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ?
=> ? = 5 + 2
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ?
=> ? = 0 + 2
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ?
=> ? = 0 + 2
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18,19],[20,21,22],[23]]
=> ?
=> ? = 4 + 2
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ?
=> ? = 5 + 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000007
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000007: Permutations ⟶ ℤResult quality: 62% values known / values provided: 94%distinct values known / distinct values provided: 62%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 0 + 3
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4 = 1 + 3
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3 = 0 + 3
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3 = 0 + 3
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3 = 0 + 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3 = 0 + 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3 = 0 + 3
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5 = 2 + 3
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 4 = 1 + 3
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3 = 0 + 3
([(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 3 = 0 + 3
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6 = 3 + 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 5 = 2 + 3
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 5 = 2 + 3
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 5 = 2 + 3
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 5 = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 5 = 2 + 3
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? => ? = 1 + 3
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [12,9,10,11,6,7,8,1,2,3,4,5] => ? = 1 + 3
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21,22,23],[24]]
=> ? => ? = 1 + 3
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 3 + 3
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24],[25]]
=> ? => ? = 4 + 3
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15,16],[17,18,19],[20]]
=> ? => ? = 4 + 3
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ? => ? = 6 + 3
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18,19,20],[21,22,23,24,25,26,27],[28,29,30,31,32],[33,34,35],[36]]
=> ? => ? = 9 + 3
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 2 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? => ? = 4 + 3
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 2 + 3
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3 + 3
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 3
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1 + 3
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? => ? = 4 + 3
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 3 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ? => ? = 5 + 3
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 0 + 3
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? => ? = 0 + 3
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? => ? = 1 + 3
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? => ? = 0 + 3
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? => ? = 0 + 3
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ? => ? = 0 + 3
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? => ? = 0 + 3
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
=> ? => ? = 0 + 3
([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> [9,6,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19]]
=> ? => ? = 0 + 3
([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> [10,6,4]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16],[17,18,19,20]]
=> ? => ? = 0 + 3
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? = 0 + 3
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ? => ? = 0 + 3
([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18],[19,20]]
=> ? => ? = 2 + 3
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? => ? = 0 + 3
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
=> ? => ? = 0 + 3
([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> [9,6,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18]]
=> ? => ? = 0 + 3
([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> [10,7,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20]]
=> ? => ? = 0 + 3
([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> [11,7,3]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21]]
=> ? => ? = 0 + 3
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ? => ? = 0 + 3
([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> [8,5,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16,17],[18,19]]
=> ? => ? = 2 + 3
([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> [9,6,5,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19,20],[21,22,23]]
=> ? => ? = 3 + 3
([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> [10,7,5,3]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17],[18,19,20,21,22],[23,24,25]]
=> ? => ? = 3 + 3
([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> [9,6,4,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15],[16,17,18,19],[20,21,22]]
=> ? => ? = 3 + 3
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? => ? = 0 + 3
([(0,7),(0,8),(1,10),(1,16),(2,11),(3,10),(4,12),(4,13),(5,3),(6,2),(6,16),(7,9),(8,5),(9,1),(9,6),(10,14),(11,12),(11,15),(12,17),(13,17),(14,13),(14,15),(15,17),(16,4),(16,11),(16,14)],18)
=> [8,6,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18]]
=> ? => ? = 0 + 3
([(0,8),(0,9),(1,15),(1,18),(2,13),(3,11),(3,17),(4,11),(5,12),(6,4),(7,5),(7,17),(8,10),(9,6),(10,3),(10,7),(11,14),(12,16),(12,18),(14,15),(14,16),(15,19),(16,19),(17,1),(17,12),(17,14),(18,2),(18,19),(19,13)],20)
=> [9,7,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20]]
=> ? => ? = 0 + 3
([(0,10),(0,12),(1,23),(2,22),(3,14),(3,24),(4,15),(5,13),(5,14),(6,18),(7,16),(7,20),(8,5),(8,23),(9,4),(9,24),(10,11),(11,3),(11,9),(12,1),(12,8),(13,22),(14,19),(15,16),(15,21),(16,25),(18,17),(19,20),(19,21),(20,18),(20,25),(21,25),(22,6),(23,2),(23,13),(24,7),(24,15),(24,19),(25,17)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ? => ? = 5 + 3
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? => ? = 0 + 3
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ? => ? = 0 + 3
([(0,9),(0,11),(1,14),(2,12),(2,13),(3,12),(3,17),(4,18),(5,15),(5,16),(6,7),(7,4),(7,13),(8,5),(8,19),(9,6),(10,2),(10,3),(10,14),(11,1),(11,10),(12,20),(13,18),(13,20),(14,8),(14,17),(15,22),(16,22),(17,19),(18,15),(18,21),(19,16),(20,21),(21,22)],23)
=> [8,6,5,3,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17,18,19],[20,21,22],[23]]
=> ? => ? = 4 + 3
([(0,10),(0,12),(1,15),(2,13),(2,14),(3,16),(3,18),(4,20),(5,13),(5,17),(6,21),(7,8),(8,6),(8,14),(9,3),(9,24),(10,7),(11,2),(11,5),(11,15),(12,1),(12,11),(13,22),(14,21),(14,22),(15,9),(15,17),(16,20),(16,25),(17,24),(18,25),(20,19),(21,18),(21,23),(22,23),(23,25),(24,4),(24,16),(25,19)],26)
=> [9,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26]]
=> ? => ? = 5 + 3
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern ([1], {(1,1)}), i.e., the upper right quadrant is shaded, see [1].
The following 518 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000507The number of ascents of a standard tableau. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000809The reduced reflection length of the permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001462The number of factors of a standard tableaux under concatenation. St001777The number of weak descents in an integer composition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000172The Grundy number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000925The number of topologically connected components of a set partition. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000006The dinv of a Dyck path. St000010The length of the partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length of a permutation. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001530The depth of a Dyck path. St000147The largest part of an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000741The Colin de Verdière graph invariant. St001152The number of pairs with even minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001568The smallest positive integer that does not appear twice in the partition. St001330The hat guessing number of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001706The number of closed sets in a graph. St000089The absolute variation of a composition. St000377The dinv defect of an integer partition. St000934The 2-degree of an integer partition. St000008The major index of the composition. St000021The number of descents of a permutation. St000047The number of standard immaculate tableaux of a given shape. St000145The Dyson rank of a partition. St000277The number of ribbon shaped standard tableaux. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001323The independence gap of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001489The maximum of the number of descents and the number of inverse descents. St001917The order of toric promotion on the set of labellings of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000470The number of runs in a permutation. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000479The Ramsey number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000553The number of blocks of a graph. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000770The major index of an integer partition when read from bottom to top. St000775The multiplicity of the largest eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000806The semiperimeter of the associated bargraph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St001812The biclique partition number of a graph. St001834The number of non-isomorphic minors of a graph. St001117The game chromatic index of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000759The smallest missing part in an integer partition. St000475The number of parts equal to 1 in a partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000913The number of ways to refine the partition into singletons. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001561The value of the elementary symmetric function evaluated at 1. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001730The number of times the path corresponding to a binary word crosses the base line. St000847The number of standard Young tableaux whose descent set is the binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000386The number of factors DDU in a Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(x^n). St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000660The number of rises of length at least 3 of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000527The width of the poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000296The length of the symmetric border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000295The length of the border of a binary word. St000627The exponent of a binary word. St001884The number of borders of a binary word. St000658The number of rises of length 2 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St000481The number of upper covers of a partition in dominance order. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St000052The number of valleys of a Dyck path not on the x-axis. St000617The number of global maxima of a Dyck path. St000306The bounce count of a Dyck path. St000655The length of the minimal rise of a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000480The number of lower covers of a partition in dominance order. St000897The number of different multiplicities of parts of an integer partition. St000920The logarithmic height of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000013The height of a Dyck path. St001696The natural major index of a standard Young tableau. St000661The number of rises of length 3 of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000340The number of non-final maximal constant sub-paths of length greater than one. St000442The maximal area to the right of an up step of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St001732The number of peaks visible from the left. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000439The position of the first down step of a Dyck path. St000632The jump number of the poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000057The Shynar inversion number of a standard tableau. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001471The magnitude of a Dyck path. St000369The dinv deficit of a Dyck path. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000932The number of occurrences of the pattern UDU in a Dyck path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000024The number of double up and double down steps of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001196The global dimension of A minus the global dimension of eAe for the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001197The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000686The finitistic dominant dimension of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n-1}] such that n=c_0 < c_i for all i > 0 a Dyck path as follows: St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001733The number of weak left to right maxima of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000256The number of parts from which one can substract 2 and still get an integer partition. St000225Difference between largest and smallest parts in a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001214The aft of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n−1}] such that n=c_0 < c_i for all i > 0 a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001194The injective dimension of A/AfA in the corresponding Nakayama algebra A when Af is the minimal faithful projective-injective left A-module St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001256Number of simple reflexive modules that are 2-stable reflexive. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St000951The dimension of Ext^{1}(D(A),A) of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001193The dimension of Ext_A^1(A/AeA,A) in the corresponding Nakayama algebra A such that eA is a minimal faithful projective-injective module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000079The number of alternating sign matrices for a given Dyck path. St000335The difference of lower and upper interactions. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000954Number of times the corresponding LNakayama algebra has Ext^i(D(A),A)=0 for i>0. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c_0,c_1,...,c_{n−1}] by adding c_0 to c_{n−1}. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules S with grade \inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \} at least two in the Nakayama algebra A corresponding to the Dyck path. St001191Number of simple modules S with Ext_A^i(S,A)=0 for all i=0,1,...,g-1 in the corresponding Nakayama algebra A with global dimension g. St001192The maximal dimension of Ext_A^2(S,A) for a simple module S over the corresponding Nakayama algebra A. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n−1}] such that n=c_0 < c_i for all i > 0 a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001481The minimal height of a peak of a Dyck path. St000443The number of long tunnels of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has Ext^i(D(A),A)>0 for i>0 for the corresponding LNakayama algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c_0,c_1,...,c_{n-1}] by adding c_0 to c_{n-1}. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000981The length of the longest zigzag subpath. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000715The number of semistandard Young tableaux of given shape and entries at most 3. St001960The number of descents of a permutation minus one if its first entry is not one. St000478Another weight of a partition according to Alladi. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St000120The number of left tunnels of a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000117The number of centered tunnels of a Dyck path. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between e_i J and e_j J (the radical of the indecomposable projective modules). St001480The number of simple summands of the module J^2/J^3. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001959The product of the heights of the peaks of a Dyck path. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000993The multiplicity of the largest part of an integer partition. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St000005The bounce statistic of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of projdim(S)+injdim(S) over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000144The pyramid weight of the Dyck path. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph.