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Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St000081
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([],5)
=> ([],5)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
Description
The number of edges of a graph.
Matching statistic: St000228
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00074: Posets —to graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Values
([],1)
=> ([],1)
=> []
=> 0
([],2)
=> ([],2)
=> []
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],3)
=> []
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([],4)
=> ([],4)
=> []
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([],5)
=> ([],5)
=> []
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000459
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00074: Posets —to graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Values
([],1)
=> ([],1)
=> []
=> 0
([],2)
=> ([],2)
=> []
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],3)
=> []
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([],4)
=> ([],4)
=> []
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([],5)
=> ([],5)
=> []
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St001251
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 96%●distinct values known / distinct values provided: 45%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
Half the sum of the even parts of a partition.
Matching statistic: St000142
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 95%●distinct values known / distinct values provided: 45%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 95%●distinct values known / distinct values provided: 45%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,4),(1,5),(2,3),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The number of even parts of a partition.
Matching statistic: St000935
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 95%●distinct values known / distinct values provided: 45%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 95%●distinct values known / distinct values provided: 45%
Values
([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> ([],2)
=> [1,1]
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
([],3)
=> ([],3)
=> [1,1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 3 = 2 + 1
([],4)
=> ([],4)
=> [1,1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 4 = 3 + 1
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 6 = 5 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 7 = 6 + 1
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 5 = 4 + 1
([(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,4),(1,5),(2,3),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> [2,2,2,2,2,2,2,1]
=> ? = 7 + 1
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10 + 1
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12 + 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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21 + 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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16 + 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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34 + 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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20 + 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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36 + 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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30 + 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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30 + 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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55 + 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48 + 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60 + 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10 + 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12 + 1
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12 + 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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20 + 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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30 + 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16 + 1
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16 + 1
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12 + 1
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14 + 1
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10 + 1
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14 + 1
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6 + 1
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4 + 1
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10 + 1
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8 + 1
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14 + 1
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10 + 1
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8 + 1
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10 + 1
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8 + 1
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12 + 1
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10 + 1
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9 + 1
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9 + 1
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9 + 1
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9 + 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St000460
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00074: Posets —to graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 95%●distinct values known / distinct values provided: 41%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 95%●distinct values known / distinct values provided: 41%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([],2)
=> []
=> ? = 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],3)
=> []
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([],4)
=> ([],4)
=> []
=> ? = 0
([(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([],5)
=> ([],5)
=> []
=> ? = 0
([(3,4)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([],6)
=> ([],6)
=> []
=> ? = 0
([],7)
=> ([],7)
=> []
=> ? = 0
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00074: Posets —to graph⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 95%●distinct values known / distinct values provided: 41%
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 95%●distinct values known / distinct values provided: 41%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([],2)
=> []
=> ? = 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],3)
=> []
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [1,1]
=> 2
([],4)
=> ([],4)
=> []
=> ? = 0
([(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([],5)
=> ([],5)
=> []
=> ? = 0
([(3,4)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([],6)
=> ([],6)
=> []
=> ? = 0
([],7)
=> ([],7)
=> []
=> ? = 0
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
([(0,7),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(1,7),(2,6),(4,2),(4,7),(5,1),(5,6)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 9
([(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7),(6,2),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001280
Mp00074: Posets —to graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 94%●distinct values known / distinct values provided: 41%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 94%●distinct values known / distinct values provided: 41%
Values
([],1)
=> ([],1)
=> [1]
=> 0
([],2)
=> ([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([],4)
=> ([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,3),(2,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(1,3),(1,4),(4,2)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 6
([(2,3),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 4
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(5,4),(6,3)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,4),(0,5),(0,6),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,4),(0,6),(1,3),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> [2,2,2,2,2,2,2,2,2]
=> ? = 9
([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ?
=> ? = 10
([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> ?
=> ? = 21
([(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)
=> ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> ?
=> ? = 16
([(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)
=> ([(0,15),(1,16),(2,9),(3,15),(3,22),(4,16),(4,22),(5,17),(5,19),(6,12),(6,17),(7,9),(7,12),(8,13),(8,18),(10,18),(10,19),(10,22),(11,20),(11,21),(11,23),(12,14),(13,14),(13,23),(14,17),(15,20),(16,21),(17,23),(18,20),(18,23),(19,21),(19,23),(20,22),(21,22)],24)
=> ?
=> ? = 34
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,24),(1,15),(2,14),(3,20),(3,22),(4,21),(4,23),(5,20),(5,24),(6,21),(6,24),(7,14),(7,22),(8,15),(8,23),(9,12),(9,14),(9,22),(10,13),(10,15),(10,23),(11,12),(11,13),(11,17),(12,18),(13,19),(16,17),(16,20),(16,21),(16,24),(17,18),(17,19),(18,20),(18,22),(19,21),(19,23)],25)
=> ?
=> ? = 36
([(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)
=> ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20)
=> ?
=> ? = 30
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(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)
=> ([(0,35),(1,19),(2,18),(3,29),(3,31),(4,30),(4,32),(5,29),(5,33),(6,30),(6,34),(7,31),(7,35),(8,32),(8,35),(9,18),(9,33),(10,19),(10,34),(11,16),(11,18),(11,33),(12,17),(12,19),(12,34),(13,14),(13,15),(13,24),(14,16),(14,25),(15,17),(15,26),(16,27),(17,28),(20,21),(20,22),(20,23),(20,24),(21,25),(21,29),(21,31),(22,26),(22,30),(22,32),(23,31),(23,32),(23,35),(24,25),(24,26),(25,27),(26,28),(27,29),(27,33),(28,30),(28,34)],36)
=> ?
=> ? = 55
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> ([(0,25),(1,24),(2,15),(3,14),(4,22),(4,28),(5,23),(5,29),(6,14),(6,22),(7,15),(7,23),(8,18),(8,20),(8,21),(9,19),(9,20),(9,21),(10,24),(10,25),(10,28),(11,24),(11,25),(11,29),(12,14),(12,20),(12,22),(13,15),(13,21),(13,23),(16,18),(16,24),(16,28),(16,29),(17,19),(17,25),(17,28),(17,29),(18,26),(18,27),(19,26),(19,27),(20,26),(21,27),(22,26),(23,27),(26,28),(27,29)],30)
=> ?
=> ? = 48
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> ([(0,28),(1,18),(2,17),(3,8),(4,15),(4,26),(5,16),(5,27),(6,17),(6,33),(7,18),(7,34),(8,10),(9,28),(9,33),(9,34),(10,15),(10,16),(11,26),(11,27),(11,35),(12,19),(12,22),(12,30),(13,20),(13,23),(13,31),(14,21),(14,24),(14,25),(15,29),(16,29),(17,22),(18,23),(19,28),(19,33),(19,35),(20,28),(20,34),(20,35),(21,29),(21,30),(21,31),(22,24),(22,33),(23,25),(23,34),(24,30),(24,32),(25,31),(25,32),(26,29),(26,30),(27,29),(27,31),(30,35),(31,35),(32,33),(32,34),(32,35)],36)
=> ?
=> ? = 60
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 12
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(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)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ?
=> ? = 20
([(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)
=> ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
=> ?
=> ? = 30
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 16
([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ?
=> ? = 12
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,3),(0,5),(1,2),(1,4),(2,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8)
=> ([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
([(0,7),(1,6),(2,5),(3,4)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ?
=> ? = 4
([(0,3),(0,6),(0,7),(1,2),(1,6),(1,7),(2,4),(3,5),(6,4),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,7),(5,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,6),(2,7),(3,4),(3,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 14
([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8)
=> ([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 10
([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ([(0,1),(2,5),(2,7),(3,4),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 8
([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1),(0,7),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 12
([(0,3),(0,6),(1,6),(1,7),(2,4),(2,5),(3,4),(3,7),(6,2),(7,5)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 10
([(0,7),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 9
Description
The number of parts of an integer partition that are at least two.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001657The number of twos in an integer partition. St001389The number of partitions of the same length below the given integer partition. St001176The size of a partition minus its first part. St000345The number of refinements of a partition. St000185The weighted size of a partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001622The number of join-irreducible elements of a lattice. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000095The number of triangles of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000327The number of cover relations in a poset. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice.
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