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Your data matches 189 different statistics following compositions of up to 3 maps.
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Matching statistic: St000128
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000128: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0
[.,[.,.]]
=> 0
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 0
[[.,.],[.,[.,.]]]
=> 0
[[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> 0
[[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> 0
[[.,[[.,.],.]],.]
=> 0
[[[.,.],[.,.]],.]
=> 0
[[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[.,[.,.]],.]]]}}} in a binary tree.
[[oeis:A159769]] counts binary trees avoiding this pattern.
Matching statistic: St000455
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1] => [1] => ([],1)
=> ? = 0
[.,[.,.]]
=> [2,1] => [2] => ([],2)
=> ? = 0
[[.,.],.]
=> [1,2] => [2] => ([],2)
=> ? = 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [3] => ([],3)
=> ? = 0
[[.,.],[.,.]]
=> [3,1,2] => [3] => ([],3)
=> ? = 0
[[.,[.,.]],.]
=> [2,1,3] => [3] => ([],3)
=> ? = 0
[[[.,.],.],.]
=> [1,2,3] => [3] => ([],3)
=> ? = 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4] => ([],4)
=> ? = 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4] => ([],4)
=> ? = 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4] => ([],4)
=> ? = 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4] => ([],4)
=> ? = 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4] => ([],4)
=> ? = 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4] => ([],4)
=> ? = 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => ([],4)
=> ? = 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5] => ([],5)
=> ? = 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5] => ([],5)
=> ? = 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5] => ([],5)
=> ? = 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5] => ([],5)
=> ? = 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [5] => ([],5)
=> ? = 0
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> 0
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [5] => ([],5)
=> ? = 0
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [5] => ([],5)
=> ? = 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [5] => ([],5)
=> ? = 0
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [5] => ([],5)
=> ? = 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [5] => ([],5)
=> ? = 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [5] => ([],5)
=> ? = 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [6,3,4,5,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [5,4,3,6,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [4,5,3,6,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [5,3,4,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [4,3,5,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [6,5,4,2,3,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [5,6,4,2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[.,[[.,.],[[[.,.],.],.]]]
=> [4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [4,5,3,2,6,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[.,[[.,[[[.,.],.],.]],.]]
=> [3,4,5,2,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[.,.],[[.,.],.]],.]]
=> [4,5,2,3,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[[.,.],.],[.,.]],.]]
=> [5,2,3,4,6,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[.,[[.,.],.]],.],.]]
=> [3,4,2,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[[.,.],[.,.]],.],.]]
=> [4,2,3,5,6,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[[.,[.,.]],.],.],.]]
=> [3,2,4,5,6,1] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[.,.],[.,[[[.,.],.],.]]]
=> [4,5,6,3,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[[.,.],[[.,.],.]]]
=> [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[[[.,.],.],[.,.]]]
=> [6,3,4,5,1,2] => [1,5] => ([(4,5)],6)
=> 0
[[.,.],[[.,[[.,.],.]],.]]
=> [4,5,3,6,1,2] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[.,.],[[[.,.],[.,.]],.]]
=> [5,3,4,6,1,2] => [1,5] => ([(4,5)],6)
=> 0
[[.,.],[[[.,[.,.]],.],.]]
=> [4,3,5,6,1,2] => [1,5] => ([(4,5)],6)
=> 0
[[.,[.,.]],[[[.,.],.],.]]
=> [4,5,6,2,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[[.,.],.],[.,[.,[.,.]]]]
=> [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[[[.,.],.],[.,[[.,.],.]]]
=> [5,6,4,1,2,3] => [2,4] => ([(3,5),(4,5)],6)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001695
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> []
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [1]
=> [[1]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1]
=> [[1]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> ? = 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 0
[.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> ? = 0
[.,[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 0
[.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 0
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> ? = 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> ? = 0
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> ? = 0
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 1
[[.,.],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> ? = 0
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Matching statistic: St001698
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> []
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [1]
=> [[1]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1]
=> [[1]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> ? = 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 0
[.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> ? = 0
[.,[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 0
[.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 0
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> ? = 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> ? = 0
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> ? = 0
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 1
[[.,.],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> ? = 0
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> []
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [1]
=> [[1]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1]
=> [[1]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> ? = 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> ? = 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> ? = 0
[.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 0
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> ? = 0
[.,[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> ? = 0
[.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> ? = 0
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> ? = 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> ? = 0
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> ? = 0
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> ? = 1
[[.,.],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> ? = 0
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> []
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [1]
=> [[1]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1]
=> [[1]]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> ? = 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> ? = 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> ? = 0
[.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> ? = 0
[.,[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 0
[.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 0
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> ? = 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> ? = 0
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
=> ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> ? = 0
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 0
[[.,.],[.,[[.,[.,.]],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> ? = 1
[[.,.],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> ? = 0
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001330
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[.,.],.]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
=> ? = 1 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 1 + 2
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[[.,.],[.,[.,[[.,.],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000741
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,.]]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [[[[],[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
=> [[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,[[[.,.],.],.]],.]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> [[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[[]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [[[[]],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [[[[],[]],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [[[],[[]],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [[[[]],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[.,[[[.,[.,.]],[.,.]],.]]
=> [[[[]],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 0 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[.,.],[.,[[.,[.,.]],.]]]
=> [[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> [[],[[],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 1
[[.,.],[[[.,.],.],[.,.]]]
=> [[],[[],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[[.,[.,[.,.]]],.]]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[.,.],[[.,[[.,.],.]],.]]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> [[],[[],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,.],[[[.,[.,.]],.],.]]
=> [[],[[[]],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[[]],[[[[]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 0 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> [[[]],[[],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[.,[.,.]],[[.,[.,.]],.]]
=> [[[]],[[[]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> [[],[],[[],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[[.,.],.],[[.,[.,.]],.]]
=> [[],[],[[[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[[[]]],[[[]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> [[],[[]],[[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001545
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[.,.],.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,.],.]]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[.,.],[.,.]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,.]],.]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,.],.],.]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[[.,[.,[[.,.],.]]],.]
=> [[[[],[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
=> [[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,[[[.,.],.],.]],.]
=> [[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> [[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[.,[.,[.,.]]],.],.]
=> [[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[[[.,[[.,.],.]],.],.]
=> [[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[[.,.],[.,.]],.],.]
=> [[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[[.,[.,.]],.],.],.]
=> [[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[[[[[.,.],.],.],.],.]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[[]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [[[[[[],[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [[[[[],[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [[[[[[]],[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [[[[[],[],[]]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [[[[],[[[]]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [[[[],[[],[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [[[[[]],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [[[[],[],[[]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [[[[[[]]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [[[[[],[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [[[[],[[]],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [[[[[]],[],[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [[[],[[[[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [[[],[[[]],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> [[[[]],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> [[[[]],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> [[[[[]]],[[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> [[[[],[]],[[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],[.,.]]]
=> [[[],[[]],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,[.,.]],.],[.,.]]]
=> [[[[]],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [[[[[[]]]],[]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[.,[[.,[[.,.],[.,.]]],.]]
=> [[[[],[[]]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[.,[[.,[.,.]],.]],.]]
=> [[[[[]],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[.,[[[.,[.,.]],[.,.]],.]]
=> [[[[]],[[]],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 0 + 2
[[.,.],[.,[.,[[.,.],.]]]]
=> [[],[[[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 2
[[.,.],[.,[[.,.],[.,.]]]]
=> [[],[[[],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[[.,.],[.,[[.,[.,.]],.]]]
=> [[],[[[[]],[]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[.,.]]]]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[[.,.],[[.,.],[[.,.],.]]]
=> [[],[[],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[.,[.,.]],[.,.]]]
=> [[],[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 0 + 2
[[.,.],[[[.,.],.],[.,.]]]
=> [[],[[],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
[[.,.],[[.,[.,[.,.]]],.]]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[[.,.],[[.,[[.,.],.]],.]]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 0 + 2
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001301
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> ([],1)
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 0
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 0
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0
[.,[[.,[[[.,.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 0
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 0
[.,[[[.,.],[[.,.],.]],.]]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 0
[.,[[[.,[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
The following 179 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000297The number of leading ones in a binary word. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000666The number of right tethers of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001890The maximum magnitude of the Möbius function of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St000096The number of spanning trees of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000287The number of connected components of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001828The Euler characteristic of a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001866The nesting alignments of a signed permutation. St000286The number of connected components of the complement of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000379The number of Hamiltonian cycles in a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000878The number of ones minus the number of zeros of a binary word. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000366The number of double descents of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001115The number of even descents of a permutation. St000383The last part of an integer composition. St000655The length of the minimal rise of a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000007The number of saliances of the permutation. St000439The position of the first down step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000629The defect of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000807The sum of the heights of the valleys of the associated bargraph. St000894The trace of an alternating sign matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000701The protection number of a binary tree. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000816The number of standard composition tableaux of the composition. St000905The number of different multiplicities of parts of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001765The number of connected components of the friends and strangers graph. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000264The girth of a graph, which is not a tree. St000822The Hadwiger number of the graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001651The Frankl number of a lattice. St001927Sparre Andersen's number of positives of a signed permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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