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Your data matches 723 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
(load all 262 compositions to match this statistic)
(load all 262 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [2,1]
=> 0
{{1},{2,3}}
=> [2,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000052
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000131
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000131: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000131: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> 0
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [[.,.],[.,[.,[[.,.],.]]]]
=> 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [[.,.],[.,[[.,.],[.,.]]]]
=> 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]]
=> 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [[.,.],[.,[[.,[.,.]],.]]]
=> 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [[.,.],[[.,.],[.,[.,.]]]]
=> 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[.,.],[[.,.],[[.,.],.]]]
=> 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [[.,.],[[.,.],[.,[.,.]]]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[[[[.,.],.],.],.]]}}} in a binary tree.
[[oeis:A159772]] counts binary trees avoiding this pattern.
Matching statistic: St000296
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [2,1]
=> 1010 => 0
{{1},{2,3}}
=> [2,1]
=> 1010 => 0
{{1,2,4},{3}}
=> [3,1]
=> 10010 => 0
{{1,2},{3,4}}
=> [2,2]
=> 1100 => 0
{{1,3,4},{2}}
=> [3,1]
=> 10010 => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> 10110 => 0
{{1},{2,3,4}}
=> [3,1]
=> 10010 => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> 10110 => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> 10110 => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> 10110 => 0
{{1,2,3,5},{4}}
=> [4,1]
=> 100010 => 0
{{1,2,3},{4,5}}
=> [3,2]
=> 10100 => 0
{{1,2,4,5},{3}}
=> [4,1]
=> 100010 => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 100110 => 0
{{1,2},{3,4,5}}
=> [3,2]
=> 10100 => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 11010 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 100110 => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 11010 => 0
{{1,3,4,5},{2}}
=> [4,1]
=> 100010 => 0
{{1,3,4},{2,5}}
=> [3,2]
=> 10100 => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 100110 => 0
{{1,3,5},{2,4}}
=> [3,2]
=> 10100 => 0
{{1,3},{2,4,5}}
=> [3,2]
=> 10100 => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 11010 => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 11010 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 101110 => 0
{{1,4},{2,3,5}}
=> [3,2]
=> 10100 => 0
{{1},{2,3,4,5}}
=> [4,1]
=> 100010 => 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> 100110 => 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 101110 => 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> 100110 => 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> 11010 => 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> 11010 => 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 101110 => 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> 11010 => 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> 100110 => 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 101110 => 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> 101110 => 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 101110 => 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> 1000010 => 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> 100100 => 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> 1000010 => 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> 1000110 => 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> 11000 => 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> 101010 => 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> 1000110 => 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> 101010 => 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> 1000010 => 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> 100100 => 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> 1000110 => 0
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 313 compositions to match this statistic)
(load all 313 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [3,2,1] => 00 => 0
{{1},{2,3}}
=> [1,3,2] => 10 => 0
{{1,2,4},{3}}
=> [2,4,3,1] => 000 => 0
{{1,2},{3,4}}
=> [2,1,4,3] => 010 => 0
{{1,3,4},{2}}
=> [3,2,4,1] => 000 => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => 001 => 0
{{1},{2,3,4}}
=> [1,3,4,2] => 100 => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => 101 => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => 000 => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 110 => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0000 => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0010 => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0000 => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0001 => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0100 => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0101 => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0000 => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0110 => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0000 => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 0000 => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0001 => 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 0000 => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0000 => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 0000 => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0010 => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0011 => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 0000 => 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1000 => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1001 => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1011 => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 0000 => 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => 0000 => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => 0000 => 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => 0001 => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 1000 => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1100 => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1101 => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 0000 => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 1110 => 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => 00000 => 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => 00010 => 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => 00000 => 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => 00001 => 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 00100 => 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => 00101 => 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => 00000 => 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => 00110 => 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => 00000 => 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => 00000 => 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => 00001 => 0
Description
The defect of a binary word.
The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St000687
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules.
This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St000768
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
St000768: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [3,2,1] => [2,1] => 0
{{1},{2,3}}
=> [1,3,2] => [1,2] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,2] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [2,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,2] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [3,1] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,2] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,1] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,2] => 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,2] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,3] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,3] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [4,1] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,3] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,3] => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,3] => 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1] => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,2] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,1] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,2] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4] => 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,4] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4] => 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,4] => 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [3,2] => 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,2] => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,3] => 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [3,2] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [2,3] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [2,3] => 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [4,1] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [3,2] => 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [5,1] => 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,2] => 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [4,2] => 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [4,2] => 0
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,3] => 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,3] => 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [5,1] => 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [4,2] => 0
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [3,3] => 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [3,3] => 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [3,3] => 0
Description
The number of peaks in an integer composition.
A peak is an ascent followed by a descent, i.e., a subsequence $c_{i-1} c_i c_{i+1}$ with $c_i > \max(c_{i-1}, c_{i+1})$.
Matching statistic: St000807
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [[1,3],[2]]
=> [2,1] => 0
{{1},{2,3}}
=> [[1,3],[2]]
=> [2,1] => 0
{{1,2,4},{3}}
=> [[1,2,4],[3]]
=> [3,1] => 0
{{1,2},{3,4}}
=> [[1,2],[3,4]]
=> [3,1] => 0
{{1,3,4},{2}}
=> [[1,3,4],[2]]
=> [2,2] => 0
{{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> [2,2] => 0
{{1},{2,3,4}}
=> [[1,3,4],[2]]
=> [2,2] => 0
{{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> [2,2] => 0
{{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> [3,1] => 0
{{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> [3,1] => 0
{{1,2,3,5},{4}}
=> [[1,2,3,5],[4]]
=> [4,1] => 0
{{1,2,3},{4,5}}
=> [[1,2,3],[4,5]]
=> [4,1] => 0
{{1,2,4,5},{3}}
=> [[1,2,4,5],[3]]
=> [3,2] => 0
{{1,2,4},{3},{5}}
=> [[1,2,4],[3],[5]]
=> [3,2] => 0
{{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> [3,2] => 0
{{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> [3,2] => 0
{{1,2,5},{3},{4}}
=> [[1,2,5],[3],[4]]
=> [4,1] => 0
{{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> [4,1] => 0
{{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> [2,3] => 0
{{1,3,4},{2,5}}
=> [[1,3,4],[2,5]]
=> [2,3] => 0
{{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> [2,3] => 0
{{1,3,5},{2,4}}
=> [[1,3,5],[2,4]]
=> [2,2,1] => 0
{{1,3},{2,4,5}}
=> [[1,3,5],[2,4]]
=> [2,2,1] => 0
{{1,3},{2,5},{4}}
=> [[1,3],[2,5],[4]]
=> [2,2,1] => 0
{{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> [2,2,1] => 0
{{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [2,3] => 0
{{1,4},{2,3,5}}
=> [[1,3,5],[2,4]]
=> [2,2,1] => 0
{{1},{2,3,4,5}}
=> [[1,3,4,5],[2]]
=> [2,3] => 0
{{1},{2,3,4},{5}}
=> [[1,3,4],[2],[5]]
=> [2,3] => 0
{{1},{2,3},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> [2,3] => 0
{{1,4,5},{2},{3}}
=> [[1,4,5],[2],[3]]
=> [3,2] => 0
{{1,4},{2,5},{3}}
=> [[1,4],[2,5],[3]]
=> [3,2] => 0
{{1,4},{2},{3,5}}
=> [[1,4],[2,5],[3]]
=> [3,2] => 0
{{1,4},{2},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> [3,2] => 0
{{1},{2,4},{3,5}}
=> [[1,4],[2,5],[3]]
=> [3,2] => 0
{{1},{2},{3,4,5}}
=> [[1,4,5],[2],[3]]
=> [3,2] => 0
{{1},{2},{3,4},{5}}
=> [[1,4],[2],[3],[5]]
=> [3,2] => 0
{{1,5},{2},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> [4,1] => 0
{{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> [4,1] => 0
{{1,2,3,4,6},{5}}
=> [[1,2,3,4,6],[5]]
=> [5,1] => 0
{{1,2,3,4},{5,6}}
=> [[1,2,3,4],[5,6]]
=> [5,1] => 0
{{1,2,3,5,6},{4}}
=> [[1,2,3,5,6],[4]]
=> [4,2] => 0
{{1,2,3,5},{4},{6}}
=> [[1,2,3,5],[4],[6]]
=> [4,2] => 0
{{1,2,3},{4,5,6}}
=> [[1,2,3],[4,5,6]]
=> [4,2] => 0
{{1,2,3},{4,5},{6}}
=> [[1,2,3],[4,5],[6]]
=> [4,2] => 0
{{1,2,3,6},{4},{5}}
=> [[1,2,3,6],[4],[5]]
=> [5,1] => 0
{{1,2,3},{4},{5,6}}
=> [[1,2,3],[4,6],[5]]
=> [5,1] => 0
{{1,2,4,5,6},{3}}
=> [[1,2,4,5,6],[3]]
=> [3,3] => 0
{{1,2,4,5},{3,6}}
=> [[1,2,4,5],[3,6]]
=> [3,3] => 0
{{1,2,4,5},{3},{6}}
=> [[1,2,4,5],[3],[6]]
=> [3,3] => 0
Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St000980
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.
Matching statistic: St001107
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
The following 713 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001172The number of 1-rises at odd height of a Dyck path. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001696The natural major index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000326The position of the first one in a binary word after appending a 1 at the end. St000659The number of rises of length at least 2 of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St000805The number of peaks of the associated bargraph. St000913The number of ways to refine the partition into singletons. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001597The Frobenius rank of a skew partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000002The number of occurrences of the pattern 123 in a permutation. St000042The number of crossings of a perfect matching. St000057The Shynar inversion number of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000210Minimum over maximum difference of elements in cycles. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000268The number of strongly connected orientations of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000347The inversion sum of a binary word. St000351The determinant of the adjacency matrix of a graph. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000386The number of factors DDU in a Dyck path. St000403The Szeged index minus the Wiener index of a graph. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. 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. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000534The number of 2-rises of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000637The length of the longest cycle in a graph. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000948The chromatic discriminant of a graph. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000974The length of the trunk of an ordered tree. St000989The number of final rises of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001091The number of parts in an integer partition whose next smaller part has the same size. St001119The length of a shortest maximal path in a graph. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001381The fertility of a permutation. St001394The genus of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001485The modular major index of a binary word. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001736The total number of cycles in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001845The number of join irreducibles minus the rank of a lattice. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000047The number of standard immaculate tableaux of a given shape. St000054The first entry of the permutation. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000115The single entry in the last row. St000160The multiplicity of the smallest part of a partition. St000183The side length of the Durfee square of an integer partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000456The monochromatic index of a connected graph. St000480The number of lower covers of a partition in dominance order. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000535The rank-width of a graph. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000627The exponent of a binary word. St000651The maximal size of a rise in a permutation. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000667The greatest common divisor of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000679The pruning number of an ordered tree. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000701The protection number of a binary tree. St000706The product of the factorials of the multiplicities of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St000733The row containing the largest entry of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000816The number of standard composition tableaux of the composition. St000823The number of unsplittable factors of the set partition. St000845The maximal number of elements covered by an element in a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000919The number of maximal left branches of a binary tree. St000920The logarithmic height of a Dyck path. St000971The smallest closer of a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001092The number of distinct even parts of a partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. 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. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001267The length of the Lyndon factorization of the binary word. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001277The degeneracy of a graph. St001280The number of parts of an integer partition that are at least two. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001344The neighbouring number of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001393The induced matching number of a graph. St001437The flex of a binary word. St001468The smallest fixpoint of a permutation. St001469The holeyness of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001546The number of monomials in the Tutte polynomial of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001587Half of the largest even part of an integer partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001592The maximal number of simple paths between any two different vertices of a graph. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001732The number of peaks visible from the left. St001743The discrepancy of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001792The arboricity of a graph. St001884The number of borders of a binary word. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000013The height of a Dyck path. St000058The order of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000258The burning number of a graph. St000397The Strahler number of a rooted tree. St000439The position of the first down step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000668The least common multiple of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000767The number of runs in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000995The largest even part of an integer partition. St001029The size of the core of a graph. St001062The maximal size of a block of a set partition. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001261The Castelnuovo-Mumford regularity of a graph. St001330The hat guessing number of a graph. St001432The order dimension of the partition. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001108The 2-dynamic chromatic number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St000264The girth of a graph, which is not a tree. St001093The detour number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001340The cardinality of a minimal non-edge isolating set of a graph. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St001518The number of graphs with the same ordinary spectrum as the given graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000962The 3-shifted major index of a permutation. St001125The number of simple modules that satisfy the 2-regular condition 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000056The decomposition (or block) number of a permutation. St000570The Edelman-Greene number of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000552The number of cut vertices of a graph. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000649The number of 3-excedences of a permutation. St000711The number of big exceedences of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001727The number of invisible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000007The number of saliances of the permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000252The number of nodes of degree 3 of a binary tree. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000834The number of right outer peaks of a permutation. 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 $c_{n−1}$. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001481The minimal height of a peak of a Dyck path. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St001434The number of negative sum pairs of a signed permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. 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. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. 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. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a 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 $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001307The number of induced stars on four vertices in a graph. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000359The number of occurrences of the pattern 23-1. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000961The shifted major index of a permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001846The number of elements which do not have a complement in the lattice. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000640The rank of the largest boolean interval in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001718The number of non-empty open intervals in a poset. St000028The number of stack-sorts needed to sort a permutation. St000647The number of big descents of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001260The permanent of an alternating sign matrix. St000124The cardinality of the preimage of the Simion-Schmidt map. St000485The length of the longest cycle of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000023The number of inner peaks of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000099The number of valleys of a permutation, including the boundary. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St000141The maximum drop size of a permutation. St000886The number of permutations with the same antidiagonal sums. St001947The number of ties in a parking function. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001734The lettericity of a graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000133The "bounce" of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000873The aix statistic of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001429The number of negative entries in a signed permutation. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. 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. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. 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. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001871The number of triconnected components of a graph. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000093The cardinality of a maximal independent set of vertices of a graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000553The number of blocks of a graph. St000652The maximal difference between successive positions of a permutation. St000654The first descent of a permutation. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St000991The number of right-to-left minima of a permutation. St001057The Grundy value of the game of creating an independent set in a graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001282The number of graphs with the same chromatic polynomial. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001512The minimum rank of a graph. St001642The Prague dimension of a graph. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001917The order of toric promotion on the set of labellings of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000062The length of the longest increasing subsequence of the permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000918The 2-limited packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001530The depth of a Dyck path. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000031The number of cycles in the cycle decomposition of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001964The interval resolution global dimension of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St001061The number of indices that are both descents and recoils of a permutation. St001430The number of positive entries in a signed permutation. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001866The nesting alignments of a signed permutation. St001948The number of augmented double ascents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001081The number of minimal length factorizations of a permutation into star transpositions. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001555The order of a signed permutation. St000454The largest eigenvalue of a graph if it is integral.
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