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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
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Mp00037: Graphs —to partition of connected components⟶ 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
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 2
([],4)
=> [1,1,1,1]
=> [4]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
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
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 98%●distinct values known / distinct values provided: 69%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 98%●distinct values known / distinct values provided: 69%
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
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
([],4)
=> [1,1,1,1]
=> [4]
=> []
=> 0
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
([],5)
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,1),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,2),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,11),(3,12),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,5),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,12),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,3),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,4),(1,5),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,3),(0,4),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,4),(1,5),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,11),(3,12),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,2),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,4),(1,5),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,2),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,12),(2,3),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000377
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
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
([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 2
([],4)
=> [1,1,1,1]
=> [3,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [4]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 3
([],5)
=> [1,1,1,1,1]
=> [3,2]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [5]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1,1,1]
=> ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> ? = 9
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1,1]
=> ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 15
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 14
([(0,1),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,2),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,11),(3,12),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,5),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,12),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,3),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,4),(1,5),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,3),(0,4),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,4),(1,5),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,11),(3,12),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,2),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,4),(1,5),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,2),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,12),(2,3),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
([(0,1),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
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
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 97%●distinct values known / distinct values provided: 69%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 97%●distinct values known / distinct values provided: 69%
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
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
([],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
([(1,3),(2,3)],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)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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
([(2,4),(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
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 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
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(2,3),(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
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? = 13
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ?
=> ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 15
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 14
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 8
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 12
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? = 12
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 7
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 9
([(5,10),(6,9),(7,8),(8,10),(9,10)],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
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? = 11
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 7
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ?
=> ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ?
=> ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ?
=> ? = 12
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 9
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 14
([(0,1),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,2),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,11),(3,12),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,1),(0,4),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,3),(1,5),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,12),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> [13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12
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)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 69% ●values known / values provided: 97%●distinct values known / distinct values provided: 69%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 69% ●values known / values provided: 97%●distinct values known / distinct values provided: 69%
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]
=> [[1,3],[2]]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 4 = 3 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5 = 4 + 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
=> ? = 13 + 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? = 9 + 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,3,4,5,6,7,8,9,10,11,12,13,14,15],[2]]
=> ? = 13 + 1
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? = 15 + 1
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? = 14 + 1
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? = 13 + 1
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? = 13 + 1
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? = 15 + 1
([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 12 + 1
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? = 14 + 1
([(0,10),(1,6),(1,10),(2,3),(2,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(7,10),(8,9)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? = 14 + 1
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 12 + 1
([(0,9),(1,5),(2,9),(2,10),(3,7),(3,10),(4,6),(4,8),(5,7),(6,9),(6,10),(7,8),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? = 13 + 1
([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 12 + 1
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? = 13 + 1
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? = 13 + 1
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? = 13 + 1
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? = 12 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
=> ? = 8 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? = 10 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
=> ? = 10 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? = 10 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
=> ? = 12 + 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? = 8 + 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
=> ? = 10 + 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? = 12 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? = 7 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
=> ? = 11 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? = 9 + 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? = 5 + 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? = 11 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? = 7 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? = 9 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
=> ? = 9 + 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? = 11 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? = 10 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? = 9 + 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? = 12 + 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? = 9 + 1
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? = 15 + 1
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? = 14 + 1
([(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,2),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,9),(3,10),(4,5),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,3),(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(1,2),(1,3),(1,5),(1,6),(1,8),(1,9),(1,10),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 1
([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,9),(2,10),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 10 + 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: St000293
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
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
([(0,2),(1,2)],3)
=> [3]
=> 1000 => 0100 => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 0100 => 2
([],4)
=> [1,1,1,1]
=> 11110 => 01111 => 0
([(2,3)],4)
=> [2,1,1]
=> 10110 => 01011 => 1
([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => 01000 => 3
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 0110 => 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => 01000 => 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 01000 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 01000 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 01000 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 01000 => 3
([],5)
=> [1,1,1,1,1]
=> 111110 => 011111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 010111 => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 01101 => 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 01010 => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 010011 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 01010 => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 010000 => 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> 10000000000000110 => ? => ? = 13
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> 1000000000110 => ? => ? = 9
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> 1000000000000010 => ? => ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> 10000000000000000 => ? => ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> 1000000000000 => ? => ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> 1000000000000000 => ? => ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> 1000000000000 => ? => ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> 100000000000000 => ? => ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> 100000000000000 => ? => ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> 10000000000000000 => ? => ? = 15
([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> [11]
=> 100000000000 => ? => ? = 10
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> 10000000000000 => ? => ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> 1000000000000000 => ? => ? = 14
([(0,10),(1,6),(1,10),(2,3),(2,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(7,10),(8,9)],11)
=> [11]
=> 100000000000 => ? => ? = 10
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> 1000000000000000 => ? => ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> 10000000000000 => ? => ? = 12
([(0,9),(1,5),(2,9),(2,10),(3,7),(3,10),(4,6),(4,8),(5,7),(6,9),(6,10),(7,8),(8,10)],11)
=> [11]
=> 100000000000 => ? => ? = 10
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> 100000000000000 => ? => ? = 13
([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> [11]
=> 100000000000 => ? => ? = 10
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> 10000000000000 => ? => ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> 100000000000000 => ? => ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> 100000000000000 => ? => ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> 100000000000000 => ? => ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> 10000000000000 => ? => ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> 1000000000000 => ? => ? = 11
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> 1000000001110 => ? => ? = 8
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> 1000000000011110 => ? => ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> 10000000000001110 => ? => ? = 12
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> 100000000110 => ? => ? = 8
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> 10000000000110 => ? => ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> 1000000000000110 => ? => ? = 12
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> 100000001110 => ? => ? = 7
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> 1000000000001110 => ? => ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 9
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> 100000000000110 => ? => ? = 11
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> 100000001110 => ? => ? = 7
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> 100000000011110 => ? => ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> 100000000000110 => ? => ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> 1000000000000110 => ? => ? = 12
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> 1000000000110 => ? => ? = 9
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> 10000000000000000 => ? => ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> 1000000000000 => ? => ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> 1000000000000000 => ? => ? = 14
([(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> 100000000000 => ? => ? = 10
([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> 100000000000 => ? => ? = 10
Description
The number of inversions of a binary word.
Matching statistic: St000074
Mp00037: Graphs —to partition of connected components⟶ 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: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
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
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 2
([],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
([(1,3),(2,3)],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)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(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),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([],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
([(2,4),(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
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 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
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,1),(2,4),(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
([(2,3),(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
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,1),(2,3),(2,4),(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
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
=> ?
=> ? = 13
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ?
=> ? = 9
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15]]
=> ?
=> ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ?
=> ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ?
=> ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ?
=> ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ?
=> ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ?
=> ? = 15
([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ?
=> ? = 10
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ?
=> ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ?
=> ? = 14
([(0,10),(1,6),(1,10),(2,3),(2,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(7,10),(8,9)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ?
=> ? = 10
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ?
=> ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ?
=> ? = 12
([(0,9),(1,5),(2,9),(2,10),(3,7),(3,10),(4,6),(4,8),(5,7),(6,9),(6,10),(7,8),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ?
=> ? = 10
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ?
=> ? = 13
([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ?
=> ? = 10
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ?
=> ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ?
=> ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ?
=> ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ?
=> ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ?
=> ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 11
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> ?
=> ? = 8
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
=> ?
=> ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15],[16]]
=> ?
=> ? = 12
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ?
=> ? = 8
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13]]
=> ?
=> ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ?
=> ? = 12
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ?
=> ? = 7
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14],[15]]
=> ?
=> ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 9
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ?
=> ? = 11
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ?
=> ? = 7
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13],[14]]
=> ?
=> ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ?
=> ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ?
=> ? = 12
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ?
=> ? = 9
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ?
=> ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[12,0,0,0,0,0,0,0,0,0,0,0],[11,0,0,0,0,0,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,0],[9,0,0,0,0,0,0,0,0],[8,0,0,0,0,0,0,0],[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ?
=> ? = 14
([(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ?
=> ? = 10
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)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
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
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
([],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
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
([],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
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
=> ?
=> ? = 13
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ?
=> ? = 9
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15]]
=> ?
=> ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ?
=> ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15]]
=> ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ?
=> ? = 15
([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15]]
=> ? = 14
([(0,10),(1,6),(1,10),(2,3),(2,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(7,10),(8,9)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15]]
=> ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 12
([(0,9),(1,5),(2,9),(2,10),(3,7),(3,10),(4,6),(4,8),(5,7),(6,9),(6,10),(7,8),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 13
([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
=> ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> [[1,10,11,12],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 8
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
=> ?
=> ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15],[16]]
=> ?
=> ? = 12
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 8
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13]]
=> ?
=> ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ?
=> ? = 12
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 7
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14],[15]]
=> ?
=> ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 9
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 5
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ?
=> ? = 11
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 7
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13],[14]]
=> ?
=> ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ?
=> ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ?
=> ? = 12
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ?
=> ? = 9
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ?
=> ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14],[15]]
=> ? = 14
([(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 10
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)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
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
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
([],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
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([],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
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
=> ? => ? = 13
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? => ? = 9
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,3,4,5,6,7,8,9,10,11,12,13,14,15],[2]]
=> ? => ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? => ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? => ? = 15
([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,10),(1,6),(1,10),(2,3),(2,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(7,10),(8,9)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,9),(1,5),(2,9),(2,10),(3,7),(3,10),(4,6),(4,8),(5,7),(6,9),(6,10),(7,8),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
=> ? => ? = 8
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? => ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
=> ? => ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? => ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
=> ? => ? = 12
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? = 8
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
=> ? => ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? => ? = 12
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 7
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
=> ? => ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? => ? = 9
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 5
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? => ? = 11
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 7
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? => ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
=> ? => ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? => ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? => ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? => ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? => ? = 12
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? => ? = 9
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? => ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
Description
The number of ascents of a permutation.
Matching statistic: St000441
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
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]
=> [[1,2],[3]]
=> [3,1,2] => 1
([(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 3
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
=> ? => ? = 13
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? => ? = 9
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15]]
=> ? => ? = 13
([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? => ? = 15
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? => ? = 15
([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,10),(1,6),(1,10),(2,3),(2,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(7,10),(8,9)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,11),(1,8),(1,14),(2,7),(2,10),(3,9),(3,10),(4,11),(4,14),(5,6),(5,11),(5,14),(6,9),(6,12),(7,8),(7,13),(8,12),(9,13),(10,13),(12,13),(12,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,9),(1,5),(2,9),(2,10),(3,7),(3,10),(4,6),(4,8),(5,7),(6,9),(6,10),(7,8),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,1),(0,2),(1,4),(2,3),(3,10),(4,11),(5,7),(5,8),(6,7),(6,9),(7,13),(8,10),(8,13),(9,11),(9,13),(10,12),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,13),(1,11),(1,12),(2,4),(2,11),(3,4),(3,10),(5,6),(5,7),(5,8),(6,10),(6,13),(7,11),(7,12),(8,12),(8,13),(9,10),(9,12),(9,13)],14)
=> [14]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
=> ? => ? = 13
([(0,12),(1,10),(1,12),(2,9),(2,12),(3,6),(3,9),(4,7),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(8,10),(8,12),(9,11),(10,11)],13)
=> [13]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? => ? = 12
([(0,1),(0,3),(1,2),(2,4),(3,5),(4,10),(5,11),(6,7),(6,8),(7,9),(8,9),(8,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> ? => ? = 8
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 10
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
=> ? => ? = 10
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 10
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15],[16]]
=> ? => ? = 12
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? = 8
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13]]
=> ? => ? = 10
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ? => ? = 12
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 7
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14],[15]]
=> ? => ? = 11
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 9
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 5
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ? => ? = 11
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 7
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 9
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13],[14]]
=> ? => ? = 9
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ? => ? = 11
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 10
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 9
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ? => ? = 12
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? => ? = 9
([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,14),(1,15),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(2,15),(3,4),(3,6),(3,7),(3,8),(3,9),(3,11),(3,13),(3,14),(3,15),(4,5),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,10),(7,12),(7,13),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(10,15),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(14,15)],16)
=> [16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]]
=> ? => ? = 15
([(0,1),(0,2),(0,3),(0,7),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,10),(1,11),(2,3),(2,5),(2,8),(2,9),(2,11),(3,4),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [1,2,3,4,5,6,7,8,9,10,11,12] => ? = 11
([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> [15]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]]
=> ? => ? = 14
([(0,3),(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 10
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000672The number of minimal elements in Bruhat order not less than the permutation. St000738The first entry in the last row of a standard tableau. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000502The number of successions of a set partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000147The largest part of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000839The largest opener of a set partition. St000369The dinv deficit of a Dyck path. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000211The rank of the set partition. St000026The position of the first return of a Dyck path. St000171The degree of the graph. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001645The pebbling number of a connected graph. St001725The harmonious chromatic number of a graph. St001330The hat guessing number of a graph. St000653The last descent of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000840The number of closers smaller than the largest opener in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001812The biclique partition number of a graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000080The rank of the poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001668The number of points of the poset minus the width of the poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation.
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