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Your data matches 220 different statistics following compositions of up to 3 maps.
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Matching statistic: St000196
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> 0
[2,1] => [[.,.],.]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> 1
[3,2,1] => [[[.,.],.],.]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 0
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 0
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 0
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 0
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000291
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 0
[2,1,3] => 10 => 1
[2,3,1] => 01 => 0
[3,1,2] => 10 => 1
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 0
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 0
[1,4,2,3] => 010 => 1
[1,4,3,2] => 011 => 0
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 0
[2,4,1,3] => 010 => 1
[2,4,3,1] => 011 => 0
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 1
[3,4,2,1] => 011 => 0
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 0
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 0
[1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => 0011 => 0
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 0
[1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => 0011 => 0
[1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0101 => 1
[1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => 0011 => 0
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 0
[2,3,1] => 01 => 1
[3,1,2] => 10 => 0
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 1
[1,4,2,3] => 010 => 1
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 0
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 1
[2,4,1,3] => 010 => 1
[2,4,3,1] => 011 => 1
[3,1,2,4] => 100 => 0
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 0
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 1
[3,4,2,1] => 011 => 1
[4,1,2,3] => 100 => 0
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 0
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 0
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => 0011 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000201
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000568
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000568: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> 1 = 0 + 1
[2,3,1] => [[.,[.,.]],.]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> 1 = 0 + 1
[3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 2 = 1 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 2 = 1 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> 3 = 2 + 1
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [1,2] => [[1,2]]
=> 0
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [1,3,2] => [[1,2],[3]]
=> 1
[2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[3,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,4,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[2,1,3,4] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[2,3,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[3,1,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[3,2,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000251
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000288
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 010 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 010 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0010 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => [1,3,4,2,5] => 0100 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0100 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0100 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0100 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0110 => 2
[1,4,3,2,5] => [1,4,2,5,3] => 0110 => 2
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0100 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0100 => 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000386
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000389
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 001 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => [1,4,2,5,3] => 0101 => 2
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0010 => 1
Description
The number of runs of ones of odd length in a binary word.
The following 210 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000390The number of runs of ones in a binary word. St000632The jump number of the poset. St000919The number of maximal left branches of a binary tree. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001214The aft of an integer partition. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000527The width of the poset. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000159The number of distinct parts of the integer partition. St000245The number of ascents of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000834The number of right outer peaks of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001176The size of a partition minus its first part. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001712The number of natural descents of a standard Young tableau. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000507The number of ascents of a standard tableau. St000522The number of 1-protected nodes of a rooted tree. St000528The height of a poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. St001732The number of peaks visible from the left. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001792The arboricity of a graph. St000354The number of recoils of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000668The least common multiple of the parts of the partition. St000884The number of isolated descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000080The rank of the poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001729The number of visible descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000353The number of inner valleys of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001668The number of points of the poset minus the width of the poset. St001737The number of descents of type 2 in a permutation. St000035The number of left outer peaks of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000711The number of big exceedences of a permutation. St001728The number of invisible descents of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000779The tier of a permutation. St001427The number of descents of a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001469The holeyness of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000374The number of exclusive right-to-left minima of a permutation. St000647The number of big descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000740The last entry of a permutation. St000646The number of big ascents of a permutation. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000523The number of 2-protected nodes of a rooted tree. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000051The size of the left subtree of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000168The number of internal nodes of an ordered tree. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000360The number of occurrences of the pattern 32-1. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001812The biclique partition number of a graph. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000409The number of pitchforks in a binary tree. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001597The Frobenius rank of a skew partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001960The number of descents of a permutation minus one if its first entry is not one. St001896The number of right descents of a signed permutations. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000640The rank of the largest boolean interval in a poset. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St001820The size of the image of the pop stack sorting operator. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. 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. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.
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