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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 0
([],2)
=> [1,1]
=> [2]
=> 0
([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> [1,1,1]
=> [3]
=> 0
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> 1
([],4)
=> [1,1,1,1]
=> [4]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(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]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([],5)
=> [1,1,1,1,1]
=> [5]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(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]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(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]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(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 size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 98%●distinct values known / distinct values provided: 73%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 98%●distinct values known / distinct values provided: 73%
Values
([],1)
=> [1]
=> [1]
=> []
=> 0
([],2)
=> [1,1]
=> [2]
=> []
=> 0
([(0,1)],2)
=> [2]
=> [1,1]
=> [1]
=> 1
([],3)
=> [1,1,1]
=> [3]
=> []
=> 0
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [4]
=> []
=> 0
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([],5)
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [2]
=> 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [2,2,2,2,1,1,1]
=> ? = 11
([(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]
=> [2,2,2,2,1,1,1,1]
=> ? = 12
([(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]
=> [3,3,2,1,1,1]
=> ? = 11
([(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]
=> [3,3,2,2,1,1,1]
=> ? = 13
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,2,2,2,1,1,1]
=> ? = 12
([(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]
=> [3,2,2,2,1,1]
=> ? = 11
([(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]
=> [2,2,2,2,1,1,1]
=> ? = 11
([(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]
=> [2,2,2,2,1,1,1,1]
=> ? = 12
([(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]
=> [3,2,2,2,1,1]
=> ? = 11
([(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]
=> [3,3,2,2,2,1,1]
=> ? = 14
([(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]
=> [3,3,2,1,1,1]
=> ? = 11
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,2,1,1]
=> ? = 14
([(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]
=> [2,2,2,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,1,1,1,1,1]
=> ? = 13
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [4,3,2,2,1,1]
=> ? = 13
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [4,3,2,2,1,1]
=> ? = 13
([(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]
=> [4,3,2,2,1,1]
=> ? = 13
([(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]
=> [4,3,2,2,1,1]
=> ? = 13
([(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]
=> [4,3,2,2,1,1]
=> ? = 13
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [4,3,2,2,1,1]
=> ? = 13
([(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]
=> [3,3,2,2,1,1]
=> ? = 12
([(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]
=> [4,4,1,1,1]
=> ? = 11
([(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]
=> [4,3,3,1,1]
=> ? = 12
([(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]
=> [4,4,1,1,1]
=> ? = 11
([(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]
=> [4,3,2,1,1]
=> ? = 11
([(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]
=> [4,3,3,1,1]
=> ? = 12
([(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]
=> [4,3,2,1,1]
=> ? = 11
([(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]
=> [4,3,2,1,1]
=> ? = 11
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000293
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 10 => 01 => 0
([],2)
=> [1,1]
=> 110 => 011 => 0
([(0,1)],2)
=> [2]
=> 100 => 010 => 1
([],3)
=> [1,1,1]
=> 1110 => 0111 => 0
([(1,2)],3)
=> [2,1]
=> 1010 => 0101 => 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1010 => 0101 => 1
([(0,2),(2,1)],3)
=> [3]
=> 1000 => 0100 => 2
([(0,2),(1,2)],3)
=> [2,1]
=> 1010 => 0101 => 1
([],4)
=> [1,1,1,1]
=> 11110 => 01111 => 0
([(2,3)],4)
=> [2,1,1]
=> 10110 => 01011 => 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> 10110 => 01011 => 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 10110 => 01011 => 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> 10110 => 01011 => 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 10110 => 01011 => 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 0110 => 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 0110 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 0110 => 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 10000 => 01000 => 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2
([],5)
=> [1,1,1,1,1]
=> 111110 => 011111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 10100 => 01010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => 01010 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> 10100 => 01010 => 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> 10100 => 01010 => 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 01101 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> 11010 => 01101 => 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]
=> 1001011100 => ? => ? = 9
([(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]
=> 1001011100 => ? => ? = 9
([(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]
=> 10010111010 => ? => ? = 9
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 1000100010 => ? => ? = 9
([(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]
=> 100001000010 => ? => ? = 12
([(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]
=> 1000101000 => ? => ? = 11
([(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]
=> 1001000010 => ? => ? = 10
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 1000100100 => ? => ? = 10
([(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]
=> 1001000100 => ? => ? = 11
([(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]
=> 1000100000 => 0100010000 => ? = 11
([(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]
=> 10000100000 => ? => ? = 12
([(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]
=> 1001000100 => ? => ? = 11
([(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]
=> 1000101000 => ? => ? = 11
([(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]
=> 1000100100 => ? => ? = 10
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 10010000010 => ? => ? = 12
([(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]
=> 100000101000 => ? => ? = 13
([(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]
=> 1001001010 => 0100100101 => ? = 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]
=> 1001001000 => ? => ? = 12
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 1001001000 => ? => ? = 12
([(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]
=> 10001011110 => 01000101111 => ? = 5
([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? = 5
([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? = 5
([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? = 5
([(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]
=> 10000001111110 => ? => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 1000010111110 => ? => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 1000010111110 => ? => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 1000010111110 => ? => ? = 6
([(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]
=> 100011011110 => ? => ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? => ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> 1000010111110 => ? => ? = 6
([(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]
=> 100011011110 => ? => ? = 6
([(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]
=> 100011011110 => ? => ? = 6
([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 100100111110 => ? => ? = 6
([(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]
=> 10101011110 => ? => ? = 6
([(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]
=> 10011100110 => ? => ? = 10
([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> [6,4,3,1]
=> 1001010010 => 0100101001 => ? = 10
([(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]
=> 1000110010 => 0100011001 => ? = 9
([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> [6,4,3,1]
=> 1001010010 => 0100101001 => ? = 10
([(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]
=> 1001100100 => 0100110010 => ? = 12
([(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]
=> 1001010100 => 0100101010 => ? = 11
([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> [6,4,3,1]
=> 1001010010 => 0100101001 => ? = 10
([(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]
=> 1000110010 => 0100011001 => ? = 9
([(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]
=> 1001100100 => 0100110010 => ? = 12
Description
The number of inversions of a binary word.
Matching statistic: St000377
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 97%●distinct values known / distinct values provided: 73%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 97%●distinct values known / distinct values provided: 73%
Values
([],1)
=> [1]
=> [1]
=> 0
([],2)
=> [1,1]
=> [2]
=> 0
([(0,1)],2)
=> [2]
=> [1,1]
=> 1
([],3)
=> [1,1,1]
=> [2,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [3]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [3]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [3]
=> 1
([],4)
=> [1,1,1,1]
=> [3,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,2]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,2]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [4]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([],5)
=> [1,1,1,1,1]
=> [3,2]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [5]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [5]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [5]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [5]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> 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]
=> ? = 12
([(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]
=> [3,3,3,3,1,1]
=> ? = 9
([(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]
=> [6,6,1]
=> ? = 8
([(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]
=> [6,6,1]
=> ? = 8
([(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]
=> [3,3,3,3,1,1]
=> ? = 9
([(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]
=> [6,6,1]
=> ? = 8
([(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]
=> [4,3,3,3,1,1]
=> ? = 9
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 11
([(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]
=> ? = 12
([(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]
=> [9,1,1,1,1,1]
=> ? = 11
([(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]
=> ? = 13
([(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]
=> ? = 10
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 10
([(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]
=> ? = 11
([(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]
=> ? = 11
([(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]
=> ? = 12
([(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]
=> ? = 11
([(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]
=> ? = 14
([(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]
=> [9,1,1,1,1,1]
=> ? = 11
([(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]
=> ? = 10
([(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]
=> ? = 12
([(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]
=> [9,1,1,1,1]
=> ? = 10
([(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]
=> ? = 14
([(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]
=> ? = 12
([(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]
=> [8,1,1,1,1,1,1,1,1]
=> ? = 13
([(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]
=> [7,2,2,1,1]
=> ? = 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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> [9,2,2,2,1,1]
=> ? = 13
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> [9,2,2,2,1,1]
=> ? = 13
([(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]
=> [9,2,2,2,1,1]
=> ? = 13
([(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]
=> [9,2,2,2,1,1]
=> ? = 13
([(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]
=> [9,2,2,2,1,1]
=> ? = 13
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> [9,2,2,2,1,1]
=> ? = 13
([(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]
=> ? = 12
([(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]
=> [4,3,2,1,1,1,1]
=> ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> [4,4,2,1,1,1]
=> ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> [4,4,2,1,1,1]
=> ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> [4,4,2,1,1,1]
=> ? = 6
([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> [4,4,2,1,1,1]
=> ? = 6
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000394
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 96%●distinct values known / distinct values provided: 87%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 96%●distinct values known / distinct values provided: 87%
Values
([],1)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(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]
=> 1
([(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]
=> 2
([(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]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(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]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(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]
=> 2
([(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]
=> 1
([(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]
=> 3
([(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]
=> 2
([],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]
=> 0
([(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]
=> 1
([(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]
=> 1
([(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]
=> 1
([(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]
=> 1
([(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]
=> 2
([(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]
=> 2
([(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]
=> 2
([(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]
=> 2
([(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]
=> 2
([(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]
=> 2
([(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]
=> 3
([(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]
=> 2
([(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]
=> 2
([(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]
=> 2
([(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]
=> 1
([(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]
=> 2
([(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]
=> 1
([(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]
=> 2
([(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]
=> 1
([(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]
=> ? = 12
([(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]
=> ? = 8
([(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]
=> ? = 10
([(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]
=> ? = 8
([(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]
=> ?
=> ? = 9
([(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]
=> ? = 8
([(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]
=> ? = 7
([(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]
=> ? = 8
([(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]
=> ? = 7
([(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]
=> ? = 7
([(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]
=> ? = 8
([(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]
=> ?
=> ? = 9
([(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]
=> ? = 8
([(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]
=> ? = 9
([(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]
=> ?
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 9
([(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]
=> ?
=> ? = 11
([(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]
=> ?
=> ? = 12
([(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]
=> ? = 11
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 10
([(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]
=> ?
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 10
([(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]
=> ?
=> ? = 14
([(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]
=> ? = 11
([(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]
=> ? = 10
([(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]
=> ?
=> ? = 12
([(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]
=> ?
=> ? = 14
([(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]
=> ?
=> ? = 12
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 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]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 12
([(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]
=> ?
=> ? = 12
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 13
([(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]
=> ? = 12
([(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]
=> ?
=> ? = 12
([(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]
=> ? = 12
([(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]
=> ?
=> ? = 13
([(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]
=> ?
=> ? = 12
([(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]
=> ? = 5
([(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 sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 96%●distinct values known / distinct values provided: 73%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 96%●distinct values known / distinct values provided: 73%
Values
([],1)
=> [1]
=> [[1]]
=> 1 = 0 + 1
([],2)
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3 = 2 + 1
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 12 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? = 9 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 9 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 1
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 12 + 1
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 12 + 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
=> ? = 9 + 1
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? = 11 + 1
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
=> ? = 12 + 1
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? = 11 + 1
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
=> ? = 13 + 1
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 8 + 1
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ? = 10 + 1
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 9 + 1
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
=> ? = 12 + 1
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ? = 10 + 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]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 9 + 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? = 11 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 9 + 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]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13]]
=> ? = 11 + 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]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14]]
=> ? = 12 + 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
=> ? = 11 + 1
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ? = 14 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
=> ? = 11 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
=> ? = 10 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 9 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 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]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 8 + 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]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 10 + 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]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 8 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 10 + 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]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
=> ? = 14 + 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]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13],[14,15,16]]
=> ? = 13 + 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]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 8 + 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]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? = 9 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 13 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
=> ? = 12 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 13 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 13 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 13 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16,17]]
=> ? = 13 + 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 12 + 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000738
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 96%●distinct values known / distinct values provided: 73%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 96%●distinct values known / distinct values provided: 73%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 1 = 0 + 1
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 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]]
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 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]]
=> 4 = 3 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,4,9,16],[2,6,12],[3,8,15],[5,11],[7,14],[10],[13]]
=> ? = 12 + 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]]
=> ? = 9 + 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]]
=> ? = 9 + 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]]
=> ? = 9 + 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]]
=> ? = 10 + 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]]
=> ? = 12 + 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]]
=> ? = 12 + 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]]
=> ? = 9 + 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]]
=> ? = 11 + 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]]
=> ? = 12 + 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]]
=> ? = 11 + 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]]
=> ? = 13 + 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]]
=> ? = 8 + 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]]
=> ? = 10 + 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]]
=> ? = 12 + 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]]
=> ? = 9 + 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]]
=> ? = 12 + 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]]
=> ? = 10 + 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]]
=> ? = 9 + 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]]
=> ? = 11 + 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]]
=> ? = 9 + 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]]
=> ? = 11 + 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]]
=> ? = 12 + 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]]
=> ? = 11 + 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]]
=> ? = 14 + 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]]
=> ? = 11 + 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]]
=> ? = 10 + 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]]
=> ? = 12 + 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]]
=> ? = 9 + 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]]
=> ? = 10 + 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]]
=> ? = 8 + 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]]
=> ? = 10 + 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]]
=> ? = 8 + 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]]
=> ? = 10 + 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]]
=> ? = 14 + 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]]
=> ? = 12 + 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]]
=> ? = 13 + 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]]
=> ? = 8 + 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]]
=> ? = 9 + 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]]
=> ? = 12 + 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]]
=> ? = 12 + 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]]
=> ? = 12 + 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]]
=> ? = 13 + 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]]
=> ? = 12 + 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]]
=> ? = 12 + 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]]
=> ? = 13 + 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]]
=> ? = 13 + 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]]
=> ? = 13 + 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]]
=> ? = 13 + 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]]
=> ? = 12 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000074
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> [[1]]
=> [[1]]
=> 0
([],2)
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
([(0,1)],2)
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> 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]]
=> ?
=> ? = 12
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[5,3,3,1,0,0,0,0,0,0,0,0],[5,3,3,0,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[5,4,3,2,1,0,0,0,0,0,0,0,0,0,0],[5,4,3,2,0,0,0,0,0,0,0,0,0,0],[5,4,3,1,0,0,0,0,0,0,0,0,0],[5,4,3,0,0,0,0,0,0,0,0,0],[5,4,2,0,0,0,0,0,0,0,0],[5,4,1,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(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]]
=> ?
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[5,3,2,2,1,0,0,0,0,0,0,0,0],[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 7
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[5,3,2,2,1,0,0,0,0,0,0,0,0],[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(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]]
=> ?
=> ? = 9
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[5,3,2,2,1,0,0,0,0,0,0,0,0],[5,3,2,2,0,0,0,0,0,0,0,0],[5,3,2,1,0,0,0,0,0,0,0],[5,3,2,0,0,0,0,0,0,0],[5,3,1,0,0,0,0,0,0],[5,3,0,0,0,0,0,0],[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 8
([(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]]
=> ?
=> ? = 9
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 13
([(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]]
=> ?
=> ? = 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 14
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 8
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 8
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 14
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 13
([(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]]
=> ?
=> ? = 8
([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[6,4,2,0,0,0,0,0,0,0,0,0],[6,4,1,0,0,0,0,0,0,0,0],[6,4,0,0,0,0,0,0,0,0],[6,3,0,0,0,0,0,0,0],[6,2,0,0,0,0,0,0],[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 9
Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> [[1]]
=> [[1]]
=> 0
([],2)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
([(0,1)],2)
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> [[1,8,13,16],[2,9,14],[3,10,15],[4,11],[5,12],[6],[7]]
=> ? = 12
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> [[1,6,9,12],[2,7,10],[3,8,11],[4],[5]]
=> ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,6,10,13,15],[2,7,11,14],[3,8,12],[4,9],[5]]
=> ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,6,9,11],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(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]]
=> ?
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> [[1,6,9,11],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,6,9,11,13],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 7
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,6,9,11,13],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(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]]
=> ?
=> ? = 9
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,6,9,11,13],[2,7,10,12],[3,8],[4],[5]]
=> ? = 8
([(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]]
=> [[1,6,9,11,13,15],[2,7,10,12,14],[3,8],[4],[5]]
=> ? = 9
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 12
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> [[1,8,13,16],[2,9,14],[3,10,15],[4,11],[5,12],[6],[7]]
=> ? = 12
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 13
([(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]]
=> ?
=> ? = 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]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 10
([(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]]
=> [[1,9],[2,10],[3,11],[4],[5],[6],[7],[8]]
=> ? = 9
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 9
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 14
([(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]]
=> ?
=> ? = 11
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 12
([(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]]
=> [[1,9],[2,10],[3,11],[4],[5],[6],[7],[8]]
=> ? = 9
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 8
([(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]]
=> ?
=> ? = 10
([(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]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 8
([(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]]
=> ?
=> ? = 10
([(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]]
=> ?
=> ? = 14
([(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]]
=> ?
=> ? = 12
([(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]]
=> ?
=> ? = 13
([(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]]
=> ?
=> ? = 8
([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
=> ? = 9
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 $i$.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> [[1]]
=> [1] => 0
([],2)
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
([(0,1)],2)
=> [2]
=> [[1,2]]
=> [1,2] => 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,3,4,8,9,15,16],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? = 12
([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11,12] => ? = 8
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10,15] => ? = 10
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,9,13,14],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 9
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11,12] => ? = 8
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12,13] => ? = 8
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 7
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 7
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12,13] => ? = 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,9,13,14],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 9
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8,12,13] => ? = 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,3,10,14,15],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? => ? = 9
([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 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]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [[1,2,3,7,8,14,15],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 12
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [[1,3,4,8,9,15,16],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? = 12
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
=> ? => ? = 9
([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [8,5,1]
=> [[1,3,4,5,6,12,13,14],[2,8,9,10,11],[7]]
=> ? => ? = 11
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [[1,3,4,5,6,12,13,14,15],[2,8,9,10,11],[7]]
=> ? => ? = 12
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [[1,2,3,7,12,13,14],[4,5,6,11],[8,9,10]]
=> ? => ? = 11
([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> [8,5,3]
=> [[1,2,3,7,8,14,15,16],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 13
([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 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]
=> [[1,3,4,5,6,12,13],[2,8,9,10,11],[7]]
=> ? => ? = 10
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [[1,2,3,7,8,14,15],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 12
([(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]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? = 9
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [[1,2,5,6,7,13,14,15],[3,4,10,11,12],[8,9]]
=> ? => ? = 12
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [[1,2,5,6,11,12,13],[3,4,9,10],[7,8]]
=> ? => ? = 10
([(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]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 9
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,5,6,7,13,14],[3,4,10,11,12],[8,9]]
=> ? => ? = 11
([(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]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? = 9
([(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]
=> [[1,2,3,4,5,11,12,13],[6,7,8,9,10]]
=> ? => ? = 11
([(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]
=> [[1,2,3,4,5,11,12,13,14],[6,7,8,9,10]]
=> ? => ? = 12
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [[1,2,5,6,7,13,14],[3,4,10,11,12],[8,9]]
=> ? => ? = 11
([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
=> [8,6,3]
=> [[1,2,3,7,8,9,16,17],[4,5,6,13,14,15],[10,11,12]]
=> ? => ? = 14
([(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]
=> [[1,2,3,7,12,13,14],[4,5,6,11],[8,9,10]]
=> ? => ? = 11
([(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]
=> [[1,2,5,6,11,12,13],[3,4,9,10],[7,8]]
=> ? => ? = 10
([(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]
=> [[1,2,3,7,8,14,15],[4,5,6,12,13],[9,10,11]]
=> ? => ? = 12
([(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]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 9
([(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]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(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]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 8
([(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]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? => ? = 10
([(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]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 8
([(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]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 10
([(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]
=> [[1,2,3,7,8,9,16,17],[4,5,6,13,14,15],[10,11,12]]
=> ? => ? = 14
([(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]
=> [[1,3,4,5,6,7,14,15],[2,9,10,11,12,13],[8]]
=> ? => ? = 12
([(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]
=> [[1,2,3,7,12,13,14,15,16],[4,5,6,11],[8,9,10]]
=> ? => ? = 13
([(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]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 8
([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? = 9
Description
The number of ascents of a permutation.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000369The dinv deficit of a Dyck path. St000502The number of successions of a set partitions. St000211The rank of the set partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000362The size of a minimal vertex cover of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001668The number of points of the poset minus the width of the poset. St001626The number of maximal proper sublattices of a lattice.
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