Your data matches 141 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [1,1]
=> 1
([(0,1)],2)
=> [2]
=> 2
([],3)
=> [1,1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> 2
([],4)
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
([],5)
=> [1,1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> 2
Description
The largest part of an integer partition.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 1
([],2)
=> [1,1]
=> [2]
=> 1
([(0,1)],2)
=> [2]
=> [1,1]
=> 2
([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> 2
Description
The length of the partition.
Matching statistic: St000378
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> [1]
=> 1
([],2)
=> [1,1]
=> [2]
=> [1,1]
=> 1
([(0,1)],2)
=> [2]
=> [1,1]
=> [2]
=> 2
([],3)
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [3]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [3]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [2,1]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [3]
=> 2
([],4)
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [4]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [4]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [4]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [3,1]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [2,2]
=> 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [3,1,1]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [2,2,1]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [2,2,1]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> [2,2,1]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [2,2,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [4,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [5]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [5]
=> 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [10,2,2]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [10,2,2]
=> ? = 5
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [4,4,3,3,1]
=> ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [4,4,2,2,2]
=> ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [7,6,2,1]
=> ? = 8
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [6,3,3,1,1,1]
=> ? = 8
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [4,4,2,2,2]
=> ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [3,2,2,2,2,2,1,1]
=> [6,3,2,2,2]
=> ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [3,3,3,2,1,1,1,1,1]
=> [7,5,2,2]
=> ? = 9
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [7,3,3,2,1,1]
=> ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [7,3,3,2,1,1]
=> ? = 7
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [7,3,3,2,1,1]
=> ? = 7
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [7,3,3,2,1,1]
=> ? = 7
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [7,3,3,2,1,1]
=> ? = 7
([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [7,3,3,2,1,1]
=> ? = 7
([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [7,6,2]
=> ? = 7
([(0,7),(1,7),(2,9),(3,10),(4,11),(5,12),(6,8),(7,12),(9,11),(10,9),(11,8),(12,10)],13)
=> [7,1,1,1,1,1,1]
=> [7,1,1,1,1,1,1]
=> [4,3,2,1,1,1,1]
=> ? = 7
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [4,4,1,1,1,1,1]
=> ? = 5
([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [4,4,1,1,1,1,1]
=> ? = 5
([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [4,4,1,1,1,1,1]
=> ? = 5
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> [4,3,2,1,1,1,1]
=> [7,3,2,1]
=> [6,2,1,1,1,1,1]
=> ? = 4
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> [5,3,3,3,1,1]
=> [6,4,4,1,1]
=> [10,3,1,1,1]
=> ? = 5
([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> [6,3,3,3,1,1]
=> [6,4,4,1,1,1]
=> [10,3,2,1,1]
=> ? = 6
([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> [6,3,3,1]
=> [4,3,3,1,1,1]
=> [4,3,2,2,2]
=> ? = 6
([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> [6,4,4,2]
=> [4,4,3,3,1,1]
=> [8,3,1,1,1,1,1]
=> ? = 6
([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> [6,3,3,3,1,1]
=> [6,4,4,1,1,1]
=> [10,3,2,1,1]
=> ? = 6
([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> [6,4,3,2]
=> [4,4,3,2,1,1]
=> [4,4,2,2,2,1]
=> ? = 6
([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> [6,3,3,1]
=> [4,3,3,1,1,1]
=> [4,3,2,2,2]
=> ? = 6
([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
=> [6,4,4,2]
=> [4,4,3,3,1,1]
=> [8,3,1,1,1,1,1]
=> ? = 6
([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> [6,4,3,2]
=> [4,4,3,2,1,1]
=> [4,4,2,2,2,1]
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> [6,4,3,2,1]
=> [5,4,3,2,1,1]
=> [9,5,2]
=> ? = 6
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000476
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St000476: Dyck paths ⟶ ℤResult quality: 70% values known / values provided: 98%distinct values known / distinct values provided: 70%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(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)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 8
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ?
=> ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 8
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 8
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 8
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 8
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 9
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ?
=> ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 8
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,1,0]
=> ?
=> ? = 8
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,1,0]
=> ?
=> ? = 9
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> [7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 10
([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
([(0,2),(0,8),(2,9),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6),(8,9)],10)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8
([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(5,9),(6,1),(7,5),(7,8),(8,9)],10)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10)
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is $$ \sum_v (j_v-i_v)/2. $$
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000288: Binary words ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 10 => 1
([],2)
=> [1,1]
=> [2]
=> 100 => 1
([(0,1)],2)
=> [2]
=> [1,1]
=> 110 => 2
([],3)
=> [1,1,1]
=> [3]
=> 1000 => 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1010 => 2
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> 1010 => 2
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 1110 => 3
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> 1010 => 2
([],4)
=> [1,1,1,1]
=> [4]
=> 10000 => 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 3
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 1100 => 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> 1100 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> 1100 => 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 11110 => 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 10100 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> 10100 => 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> 10110110110 => ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> 1100010110 => ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> 1100010110 => ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> 10100010110 => ? = 5
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> 10110110110 => ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> 1011101110 => ? = 7
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> 10111101110 => ? = 8
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> 101111011110 => ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> 1110101110 => ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [3,3,3,2,2,1,1,1]
=> 11101101110 => ? = 8
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> 101110110 => ? = 6
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> 1011110110 => ? = 7
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> 11011101110 => ? = 8
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> 1101101110 => ? = 7
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> 1101110110 => ? = 7
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> 1111101110 => ? = 8
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [2,2,2,2,2,1,1,1,1]
=> 11111011110 => ? = 9
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> 1101110110 => ? = 7
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> 1110101110 => ? = 7
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> 1101101110 => ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> 101110110 => ? = 6
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> 110101110 => ? = 6
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 8
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [3,2,2,2,2,2,1,1]
=> 10111110110 => ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [3,3,3,2,1,1,1,1,1]
=> 111010111110 => ? = 9
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> 101110110 => ? = 6
([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> 1010110110 => ? = 6
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> 10110110110 => ? = 7
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> 10110110110 => ? = 7
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> 11010110110 => ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> 11010110110 => ? = 7
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> 11010110110 => ? = 7
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> 11010110110 => ? = 7
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> 11010110110 => ? = 7
([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> 10110110110 => ? = 7
([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> 111110110 => ? = 7
([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> 10110110110 => ? = 7
([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> 11010110110 => ? = 7
([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> 1110110110 => ? = 7
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000013
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
St000013: Dyck paths ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 5
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ?
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ?
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 5
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 7
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 8
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
=> ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 8
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 7
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0,0]
=> ? = 8
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 7
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
=> ? = 7
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 8
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ?
=> ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 9
([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> [6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ?
=> ? = 6
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 7
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 7
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 7
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 7
([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 7
([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ?
=> ? = 7
([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 7
([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000734
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [[1]]
=> 1
([],2)
=> [1,1]
=> [[1],[2]]
=> 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 5
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 7
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? = 7
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? = 8
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
=> ? = 8
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ? = 7
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 7
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
=> ? = 8
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ? = 7
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 8
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? = 7
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13]]
=> ? = 8
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14]]
=> ? = 9
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? = 7
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? = 7
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 7
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 8
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 6
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 6
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 7
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ? = 8
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15]]
=> ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13],[14,15,16]]
=> ? = 9
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> [6,4,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? = 6
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 7
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 7
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 7
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 7
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 7
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 7
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 7
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 7
([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 7
Description
The last entry in the first row of a standard tableau.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 1
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12],[13],[14]]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12],[13],[14]]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14],[15]]
=> ? = 5
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 7
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 7
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12]]
=> ? = 7
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12],[13],[14]]
=> ? = 8
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12],[13],[14],[15]]
=> ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13],[14]]
=> ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15],[16]]
=> ? = 8
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 6
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12],[13]]
=> ? = 7
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 7
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13],[14],[15]]
=> ? = 8
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12],[13]]
=> ? = 7
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 8
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13],[14]]
=> ? = 7
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 7
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12],[13]]
=> ? = 8
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12],[13],[14]]
=> ? = 9
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13],[14]]
=> ? = 7
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14,15],[16],[17]]
=> ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13],[14]]
=> ? = 7
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12],[13]]
=> ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 7
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 8
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 7
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 6
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> [3,3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13]]
=> ? = 6
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 6
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 7
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14,15],[16],[17]]
=> ? = 8
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [3,2,2,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14],[15]]
=> ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [3,3,3,2,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13],[14],[15],[16]]
=> ? = 9
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 6
([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12],[13]]
=> ? = 6
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 7
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 7
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 7
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14,15],[16],[17]]
=> ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 7
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14,15],[16],[17]]
=> ? = 7
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14,15],[16],[17]]
=> ? = 7
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14,15],[16],[17]]
=> ? = 7
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14,15],[16],[17]]
=> ? = 7
([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 7
Description
The row containing the largest entry of a standard tableau.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,4,9,16],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [[1,4,7,8,9],[2,6,12,13,14],[3,11],[5],[10]]
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [[1,4,7,8,9],[2,6,12,13,14],[3,11],[5],[10]]
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [[1,4,7,8,9,15],[2,6,12,13,14],[3,11],[5],[10]]
=> ? = 5 - 1
([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 7 - 1
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,4,9,16],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> [[1,5,12],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 7 - 1
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [[1,5,14],[2,7],[3,9],[4,11],[6,13],[8],[10],[12]]
=> ? = 8 - 1
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [3,2,2,2,2,1,1,1,1]
=> [[1,6,15],[2,8],[3,10],[4,12],[5,14],[7],[9],[11],[13]]
=> ? = 9 - 1
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10,14],[4,13],[6],[9],[12]]
=> ? = 7 - 1
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [3,3,3,2,2,1,1,1]
=> [[1,5,10],[2,7,13],[3,9,16],[4,12],[6,15],[8],[11],[14]]
=> ? = 8 - 1
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 6 - 1
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [3,2,2,2,2,1,1]
=> [[1,4,13],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 7 - 1
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 7 - 1
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [[1,5,12],[2,7,15],[3,9],[4,11],[6,14],[8],[10],[13]]
=> ? = 8 - 1
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [[1,5,10],[2,7,13],[3,9],[4,12],[6],[8],[11]]
=> ? = 7 - 1
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 8 - 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8],[5,10],[7,13],[9],[12]]
=> ? = 7 - 1
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 7 - 1
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6,13],[8],[10],[12]]
=> ? = 8 - 1
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5,14],[7],[9],[11],[13]]
=> ? = 9 - 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8],[5,10],[7,13],[9],[12]]
=> ? = 7 - 1
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 8 - 1
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [3,3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10,14],[4,13],[6],[9],[12]]
=> ? = 7 - 1
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [3,3,2,2,1,1,1]
=> [[1,5,10],[2,7,13],[3,9],[4,12],[6],[8],[11]]
=> ? = 7 - 1
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 8 - 1
([(0,7),(0,8),(1,9),(2,10),(3,6),(3,9),(4,3),(5,1),(6,2),(6,11),(7,4),(8,5),(9,11),(11,10)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 7 - 1
([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 6 - 1
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> [6,4,3]
=> [3,3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? = 6 - 1
([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 6 - 1
([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 7 - 1
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> [[1,4,11],[2,6,14],[3,8,17],[5,10],[7,13],[9,16],[12],[15]]
=> ? = 8 - 1
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [3,2,2,2,2,2,1,1]
=> [[1,4,15],[2,6],[3,8],[5,10],[7,12],[9,14],[11],[13]]
=> ? = 8 - 1
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [3,3,3,2,1,1,1,1,1]
=> [[1,7,10],[2,9,13],[3,12,16],[4,15],[5],[6],[8],[11],[14]]
=> ? = 9 - 1
([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 6 - 1
([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? = 6 - 1
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,4,9,16],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,4,9,16],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12,17],[3,8,16],[5,11],[7,15],[10],[14]]
=> ? = 7 - 1
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [3,3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12,17],[3,8,16],[5,11],[7,15],[10],[14]]
=> ? = 7 - 1
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12,17],[3,8,16],[5,11],[7,15],[10],[14]]
=> ? = 7 - 1
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12,17],[3,8,16],[5,11],[7,15],[10],[14]]
=> ? = 7 - 1
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [4,4,3,2,2,1,1]
=> [[1,4,9,13],[2,6,12,17],[3,8,16],[5,11],[7,15],[10],[14]]
=> ? = 7 - 1
([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,4,9,16],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 7 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% values known / values provided: 96%distinct values known / distinct values provided: 80%
Values
([],1)
=> [1]
=> [1,0]
=> 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 5
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 7
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 8
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 9
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 8
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 8
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 7
([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 7
([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> [8,5]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 8
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 9
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 7
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 8
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 7
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 8
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 8
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 9
([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> [6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 6
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 7
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 7
([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 7
([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> [7,5,3,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 7
([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000691The number of changes of a binary word. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000025The number of initial rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000653The last descent of a permutation. St001497The position of the largest weak excedence of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000738The first entry in the last row of a standard tableau. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St000383The last part of an integer composition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001777The number of weak descents in an integer composition. St001029The size of the core of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000083The number of left oriented leafs of a binary tree except the first one. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000746The number of pairs with odd minimum in a perfect matching. St000093The cardinality of a maximal independent set of vertices of a graph. St000741The Colin de Verdière graph invariant. St000528The height of a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000308The height of the tree associated to a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000080The rank of the poset. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000926The clique-coclique number of a graph. St000991The number of right-to-left minima of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001812The biclique partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St001651The Frankl number of a lattice. St001720The minimal length of a chain of small intervals in a lattice.