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Matching statistic: St000228
(load all 2039 compositions to match this statistic)
(load all 2039 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3]
=> 3
[3,2,1] => [1,1,1]
=> 3
[1,2,3,4] => [4]
=> 4
[2,1,4,3] => [2,2]
=> 4
[3,1,4,2] => [2,2]
=> 4
[4,3,2,1] => [1,1,1,1]
=> 4
[1,2,3,4,5] => [5]
=> 5
[5,4,3,2,1] => [1,1,1,1,1]
=> 5
[1,2,3,4,5,6] => [6]
=> 6
[2,1,4,3,6,5] => [3,3]
=> 6
[2,1,5,3,6,4] => [3,3]
=> 6
[3,1,4,2,6,5] => [3,3]
=> 6
[3,1,5,2,6,4] => [3,3]
=> 6
[3,2,1,6,5,4] => [2,2,2]
=> 6
[4,1,5,2,6,3] => [3,3]
=> 6
[4,2,1,6,5,3] => [2,2,2]
=> 6
[4,3,1,6,5,2] => [2,2,2]
=> 6
[5,2,1,6,4,3] => [2,2,2]
=> 6
[5,3,1,6,4,2] => [2,2,2]
=> 6
[6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 6
[1,2,3,4,5,6,7] => [7]
=> 7
[7,6,5,4,3,2,1] => [1,1,1,1,1,1,1]
=> 7
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000926
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000926: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000926: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => ([],3)
=> 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => ([],4)
=> 4
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 4
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => ([],5)
=> 5
[5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,4,5,6] => ([],6)
=> 6
[2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> 6
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 6
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> 6
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
[3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> 6
[4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[4,2,1,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 6
[4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
[5,2,1,6,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
[5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,3,4,5,6,7] => ([],7)
=> 7
[7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
Description
The clique-coclique number of a graph.
This is the product of the size of a maximal clique [[St000097]] and the size of a maximal independent set [[St000093]].
Matching statistic: St001004
(load all 441 compositions to match this statistic)
(load all 441 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => 3
[3,2,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => 4
[2,1,4,3] => [1,4,2,3] => 4
[3,1,4,2] => [1,4,2,3] => 4
[4,3,2,1] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => 5
[5,4,3,2,1] => [1,2,3,4,5] => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[2,1,4,3,6,5] => [1,4,2,3,6,5] => 6
[2,1,5,3,6,4] => [1,5,2,3,6,4] => 6
[3,1,4,2,6,5] => [1,4,2,6,3,5] => 6
[3,1,5,2,6,4] => [1,5,2,6,3,4] => 6
[3,2,1,6,5,4] => [1,6,2,3,4,5] => 6
[4,1,5,2,6,3] => [1,5,2,6,3,4] => 6
[4,2,1,6,5,3] => [1,6,2,3,4,5] => 6
[4,3,1,6,5,2] => [1,6,2,3,4,5] => 6
[5,2,1,6,4,3] => [1,6,2,3,4,5] => 6
[5,3,1,6,4,2] => [1,6,2,3,4,5] => 6
[6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 7
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a 321 pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001020
(load all 917 compositions to match this statistic)
(load all 917 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> 6
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> 6
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> 6
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[4,1,5,2,6,3] => [1,1,1,1,0,0,1,0,0,1,0,0]
=> 6
[4,2,1,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 6
[4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> 6
[5,2,1,6,4,3] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6
[5,3,1,6,4,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6
[6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000296
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000296: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => 00 => 2 = 3 - 1
[3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => 000 => 3 = 4 - 1
[2,1,4,3] => 101 => 3 = 4 - 1
[3,1,4,2] => 101 => 3 = 4 - 1
[4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => 0000 => 4 = 5 - 1
[5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,2,3,4,5,6] => 00000 => 5 = 6 - 1
[2,1,4,3,6,5] => 10101 => 5 = 6 - 1
[2,1,5,3,6,4] => 10101 => 5 = 6 - 1
[3,1,4,2,6,5] => 10101 => 5 = 6 - 1
[3,1,5,2,6,4] => 10101 => 5 = 6 - 1
[3,2,1,6,5,4] => 11011 => 5 = 6 - 1
[4,1,5,2,6,3] => 10101 => 5 = 6 - 1
[4,2,1,6,5,3] => 11011 => 5 = 6 - 1
[4,3,1,6,5,2] => 11011 => 5 = 6 - 1
[5,2,1,6,4,3] => 11011 => 5 = 6 - 1
[5,3,1,6,4,2] => 11011 => 5 = 6 - 1
[6,5,4,3,2,1] => 11111 => 5 = 6 - 1
[1,2,3,4,5,6,7] => 000000 => 6 = 7 - 1
[7,6,5,4,3,2,1] => 111111 => 6 = 7 - 1
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: St001332
(load all 850 compositions to match this statistic)
(load all 850 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St001332: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001332: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => 2 = 3 - 1
[3,2,1] => [1,2,3] => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[2,1,4,3] => [1,4,2,3] => 3 = 4 - 1
[3,1,4,2] => [1,4,2,3] => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 6 - 1
[2,1,4,3,6,5] => [1,4,2,3,6,5] => 5 = 6 - 1
[2,1,5,3,6,4] => [1,5,2,3,6,4] => 5 = 6 - 1
[3,1,4,2,6,5] => [1,4,2,6,3,5] => 5 = 6 - 1
[3,1,5,2,6,4] => [1,5,2,6,3,4] => 5 = 6 - 1
[3,2,1,6,5,4] => [1,6,2,3,4,5] => 5 = 6 - 1
[4,1,5,2,6,3] => [1,5,2,6,3,4] => 5 = 6 - 1
[4,2,1,6,5,3] => [1,6,2,3,4,5] => 5 = 6 - 1
[4,3,1,6,5,2] => [1,6,2,3,4,5] => 5 = 6 - 1
[5,2,1,6,4,3] => [1,6,2,3,4,5] => 5 = 6 - 1
[5,3,1,6,4,2] => [1,6,2,3,4,5] => 5 = 6 - 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => 5 = 6 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 6 = 7 - 1
Description
The number of steps on the non-negative side of the walk associated with the permutation.
Consider the walk taking an up step for each ascent, and a down step for each descent of the permutation. Then this statistic is the number of steps that begin and end at non-negative height.
Matching statistic: St001415
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St001415: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001415: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => 00 => 2 = 3 - 1
[3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => 000 => 3 = 4 - 1
[2,1,4,3] => 101 => 3 = 4 - 1
[3,1,4,2] => 101 => 3 = 4 - 1
[4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => 0000 => 4 = 5 - 1
[5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,2,3,4,5,6] => 00000 => 5 = 6 - 1
[2,1,4,3,6,5] => 10101 => 5 = 6 - 1
[2,1,5,3,6,4] => 10101 => 5 = 6 - 1
[3,1,4,2,6,5] => 10101 => 5 = 6 - 1
[3,1,5,2,6,4] => 10101 => 5 = 6 - 1
[3,2,1,6,5,4] => 11011 => 5 = 6 - 1
[4,1,5,2,6,3] => 10101 => 5 = 6 - 1
[4,2,1,6,5,3] => 11011 => 5 = 6 - 1
[4,3,1,6,5,2] => 11011 => 5 = 6 - 1
[5,2,1,6,4,3] => 11011 => 5 = 6 - 1
[5,3,1,6,4,2] => 11011 => 5 = 6 - 1
[6,5,4,3,2,1] => 11111 => 5 = 6 - 1
[1,2,3,4,5,6,7] => 000000 => 6 = 7 - 1
[7,6,5,4,3,2,1] => 111111 => 6 = 7 - 1
Description
The length of the longest palindromic prefix of a binary word.
Matching statistic: St001416
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St001416: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001416: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => 00 => 2 = 3 - 1
[3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => 000 => 3 = 4 - 1
[2,1,4,3] => 101 => 3 = 4 - 1
[3,1,4,2] => 101 => 3 = 4 - 1
[4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => 0000 => 4 = 5 - 1
[5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,2,3,4,5,6] => 00000 => 5 = 6 - 1
[2,1,4,3,6,5] => 10101 => 5 = 6 - 1
[2,1,5,3,6,4] => 10101 => 5 = 6 - 1
[3,1,4,2,6,5] => 10101 => 5 = 6 - 1
[3,1,5,2,6,4] => 10101 => 5 = 6 - 1
[3,2,1,6,5,4] => 11011 => 5 = 6 - 1
[4,1,5,2,6,3] => 10101 => 5 = 6 - 1
[4,2,1,6,5,3] => 11011 => 5 = 6 - 1
[4,3,1,6,5,2] => 11011 => 5 = 6 - 1
[5,2,1,6,4,3] => 11011 => 5 = 6 - 1
[5,3,1,6,4,2] => 11011 => 5 = 6 - 1
[6,5,4,3,2,1] => 11111 => 5 = 6 - 1
[1,2,3,4,5,6,7] => 000000 => 6 = 7 - 1
[7,6,5,4,3,2,1] => 111111 => 6 = 7 - 1
Description
The length of a longest palindromic factor of a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.
Matching statistic: St001417
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => 00 => 2 = 3 - 1
[3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => 000 => 3 = 4 - 1
[2,1,4,3] => 101 => 3 = 4 - 1
[3,1,4,2] => 101 => 3 = 4 - 1
[4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => 0000 => 4 = 5 - 1
[5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,2,3,4,5,6] => 00000 => 5 = 6 - 1
[2,1,4,3,6,5] => 10101 => 5 = 6 - 1
[2,1,5,3,6,4] => 10101 => 5 = 6 - 1
[3,1,4,2,6,5] => 10101 => 5 = 6 - 1
[3,1,5,2,6,4] => 10101 => 5 = 6 - 1
[3,2,1,6,5,4] => 11011 => 5 = 6 - 1
[4,1,5,2,6,3] => 10101 => 5 = 6 - 1
[4,2,1,6,5,3] => 11011 => 5 = 6 - 1
[4,3,1,6,5,2] => 11011 => 5 = 6 - 1
[5,2,1,6,4,3] => 11011 => 5 = 6 - 1
[5,3,1,6,4,2] => 11011 => 5 = 6 - 1
[6,5,4,3,2,1] => 11111 => 5 = 6 - 1
[1,2,3,4,5,6,7] => 000000 => 6 = 7 - 1
[7,6,5,4,3,2,1] => 111111 => 6 = 7 - 1
Description
The length of a longest palindromic subword of a binary word.
A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword.
Any binary word of length n contains a palindromic subword of length at least n/2, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with n/2 zeros and ending with n/2 ones.
Matching statistic: St000054
(load all 109 compositions to match this statistic)
(load all 109 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [3,2,1] => 3
[3,2,1] => [1,2,3] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => 4
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => 4
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[2,1,4,3,6,5] => [1,4,2,3,6,5] => [6,3,5,4,1,2] => 6
[2,1,5,3,6,4] => [1,5,2,3,6,4] => [6,2,5,4,1,3] => 6
[3,1,4,2,6,5] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => 6
[3,1,5,2,6,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => 6
[3,2,1,6,5,4] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 6
[4,1,5,2,6,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => 6
[4,2,1,6,5,3] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 6
[4,3,1,6,5,2] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 6
[5,2,1,6,4,3] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 6
[5,3,1,6,4,2] => [1,6,2,3,4,5] => [6,1,5,4,3,2] => 6
[6,5,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 7
[7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 7
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. Then, \pi(1)-1 is the number of rotations of \pi where the arc (1, \pi(1)) is a deficiency. In particular, if O(\pi) is the orbit of rotations of \pi, then the number of deficiencies of \pi equals
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
The following 734 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. St000294The number of distinct factors of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000518The number of distinct subsequences in a binary word. St000548The number of different non-empty partial sums of an integer partition. St000636The hull number of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000722The number of different neighbourhoods in a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000806The semiperimeter of the associated bargraph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001034The area of the parallelogram polyomino associated with the Dyck path. 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. St001342The number of vertices in the center of a graph. St001497The position of the largest weak excedence of a permutation. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001622The number of join-irreducible elements of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001746The coalition number of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St000019The cardinality of the support of a permutation. St000081The number of edges of a graph. St000141The maximum drop size of a permutation. St000189The number of elements in the poset. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000543The size of the conjugacy class of a binary word. St000553The number of blocks of a graph. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001430The number of positive entries in a signed permutation. St001437The flex of a binary word. St001479The number of bridges of a graph. St001884The number of borders of a binary word. St001917The order of toric promotion on the set of labellings of a graph. St000295The length of the border of a binary word. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000519The largest length of a factor maximising the subword complexity. St000653The last descent of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000007The number of saliances of the permutation. St000018The number of inversions of a permutation. St000022The number of fixed points of a permutation. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000058The order of a permutation. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000203The number of external nodes of a binary tree. St000215The number of adjacencies of a permutation, zero appended. St000246The number of non-inversions of a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000271The chromatic index of a graph. St000290The major index of a binary word. St000308The height of the tree associated to a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000461The rix statistic of a permutation. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000503The maximal difference between two elements in a common block. St000505The biggest entry in the block containing the 1. St000520The number of patterns in a permutation. St000528The height of a poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000625The sum of the minimal distances to a greater element. St000654The first descent of a permutation. St000657The smallest part of an integer composition. St000702The number of weak deficiencies of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000784The maximum of the length and the largest part of the integer partition. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St000863The length of the first row of the shifted shape of a permutation. St000873The aix statistic of a permutation. St000883The number of longest increasing subsequences of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001074The number of inversions of the cyclic embedding of a permutation. St001093The detour number of a graph. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001268The size of the largest ordinal summand in the poset. St001279The sum of the parts of an integer partition that are at least two. St001343The dimension of the reduced incidence algebra of a poset. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001439The number of even weak deficiencies and of odd weak exceedences. St001461The number of topologically connected components of the chord diagram of a permutation. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001566The length of the longest arithmetic progression in a permutation. St001652The length of a longest interval of consecutive numbers. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000028The number of stack-sorts needed to sort a permutation. St000050The depth or height of a binary tree. St000063The number of linear extensions of a certain poset defined for an integer partition. St000064The number of one-box pattern of a permutation. St000070The number of antichains in a poset. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000171The degree of the graph. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000259The diameter of a connected graph. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000363The number of minimal vertex covers of a graph. St000441The number of successions of a permutation. St000501The size of the first part in the decomposition of a permutation. St000529The number of permutations whose descent word is the given binary word. St000532The total number of rook placements on a Ferrers board. St000546The number of global descents of a permutation. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000696The number of cycles in the breakpoint graph of a permutation. St000730The maximal arc length of a set partition. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000744The length of the path to the largest entry in a standard Young tableau. St000770The major index of an integer partition when read from bottom to top. St000778The metric dimension of a graph. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000844The size of the largest block in the direct sum decomposition of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. 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. St000890The number of nonzero entries in an alternating sign matrix. St000921The number of internal inversions of a binary word. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St000989The number of final rises of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001400The total number of Littlewood-Richardson tableaux of given shape. St001405The number of bonds in a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001527The cyclic permutation representation number of an integer partition. St001614The cyclic permutation representation number of a skew partition. St001631The number of simple modules S with dim Ext^1(S,A)=1 in the incidence algebra A of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001777The number of weak descents in an integer composition. St001780The order of promotion on the set of standard tableaux of given shape. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001883The mutual visibility number of a graph. St001949The rigidity index of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000024The number of double up and double down steps of a Dyck path. St000060The greater neighbor of the maximum. St000088The row sums of the character table of the symmetric group. St000148The number of odd parts of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000209Maximum difference of elements in cycles. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000277The number of ribbon shaped standard tableaux. St000313The number of degree 2 vertices of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000362The size of a minimal vertex cover of a graph. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000552The number of cut vertices of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000656The number of cuts of a poset. St000691The number of changes of a binary word. St000837The number of ascents of distance 2 of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001176The size of a partition minus its first part. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001308The number of induced paths on three vertices in a graph. St001368The number of vertices of maximal degree in a graph. St001391The disjunction number of a graph. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001521Half the total irregularity of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001692The number of vertices with higher degree than the average degree in a graph. 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. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001814The number of partitions interlacing the given partition. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St000447The number of pairs of vertices of a graph with distance 3. St000829The Ulam distance of a permutation to the identity permutation. St001306The number of induced paths on four vertices in a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000110The number of permutations less than or equal to a permutation in left weak order. St000163The size of the orbit of the set partition under rotation. St001519The pinnacle sum of a permutation. St000010The length of the partition. St000924The number of topologically connected components of a perfect matching. St001664The number of non-isomorphic subposets of a poset. St000809The reduced reflection length of the permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000197The number of entries equal to positive one in the alternating sign matrix. St000619The number of cyclic descents of a permutation. St001516The number of cyclic bonds of a permutation. St000242The number of indices that are not cyclical small weak excedances. St001489The maximum of the number of descents and the number of inverse descents. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000087The number of induced subgraphs. St000213The number of weak exceedances (also weak excedences) of a permutation. St000221The number of strong fixed points of a permutation. St000226The convexity of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000470The number of runs in a permutation. St000471The sum of the ascent tops of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000632The jump number of the poset. St000673The number of non-fixed points of a permutation. St000680The Grundy value for Hackendot on posets. St000703The number of deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000733The row containing the largest entry of a standard tableau. St000906The length of the shortest maximal chain in a poset. St000947The major index east count of a Dyck path. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001118The acyclic chromatic index of a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of projdim(S)+injdim(S) over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001304The number of maximally independent sets of vertices of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001717The largest size of an interval in a poset. St001806The upper middle entry of a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000312The number of leaves in a graph. St000354The number of recoils of a permutation. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000456The monochromatic index of a connected graph. St000527The width of the poset. St000530The number of permutations with the same descent word as the given permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000652The maximal difference between successive positions of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000864The number of circled entries of the shifted recording tableau of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001077The prefix exchange distance of a permutation. St001117The game chromatic index of a graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001468The smallest fixpoint of a permutation. St001480The number of simple summands of the module J^2/J^3. St001649The length of a longest trail in a graph. St001782The order of rowmotion on the set of order ideals of a poset. St001963The tree-depth of a graph. St000304The load of a permutation. St000309The number of vertices with even degree. St000327The number of cover relations in a poset. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000836The number of descents of distance 2 of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001160The number of proper blocks (or intervals) of a permutations. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001557The number of inversions of the second entry 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. St001727The number of invisible inversions of a permutation. St001759The Rajchgot index of a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001911A descent variant minus the number of inversions. St001812The biclique partition number of a graph. St001959The product of the heights of the peaks of a Dyck path. St000005The bounce statistic of a Dyck path. St000029The depth of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001645The pebbling number 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000719The number of alignments in a perfect matching. St001684The reduced word complexity of a permutation. St000225Difference between largest and smallest parts in a partition. 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. St001060The distinguishing index of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001427The number of descents of a signed permutation. St001651The Frankl number of a lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000668The least common multiple of the parts of the partition. St001616The number of neutral elements in a lattice. St001626The number of maximal proper sublattices of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001875The number of simple modules with projective dimension at most 1. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001820The size of the image of the pop stack sorting operator. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001623The number of doubly irreducible elements of a lattice. St001846The number of elements which do not have a complement in the lattice. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000273The domination number of a graph. St000482The (zero)-forcing number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001829The common independence number of a graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001340The cardinality of a minimal non-edge isolating set of a graph. St001175The size of a partition minus the hook length of the base cell. St000258The burning number of a graph. St000544The cop number of a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001286The annihilation number of a graph. St001363The Euler characteristic of a graph according to Knill. St000172The Grundy number of a graph. St000917The open packing number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001644The dimension of a graph. St000178Number of free entries. St000264The girth of a graph, which is not a tree. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000287The number of connected components of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001828The Euler characteristic of a graph. St000286The number of connected components of the complement of a graph. St001316The domatic number of a graph. St001401The number of distinct entries in a semistandard tableau. St001861The number of Bruhat lower covers of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001845The number of join irreducibles minus the rank of a lattice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St001556The number of inversions of the third entry of a permutation. St001429The number of negative entries in a signed permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St000135The number of lucky cars of the parking function. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001927Sparre Andersen's number of positives of a signed permutation. St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St000044The number of vertices of the unicellular map given by a perfect matching. St000067The inversion number of the alternating sign matrix. St000134The size of the orbit of an alternating sign matrix under gyration. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000315The number of isolated vertices of a graph. St000446The disorder of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000796The stat' of a permutation. St000798The makl of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001345The Hamming dimension of a graph. St001428The number of B-inversions of a signed permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. 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. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St000095The number of triangles of a graph. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000231Sum of the maximal elements of the blocks of a set partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000741The Colin de Verdière graph invariant. St000794The mak of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000909The number of maximal chains of maximal size in a poset. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001742The difference of the maximal and the minimal degree in a graph. St001948The number of augmented double ascents of a permutation. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. St001811The Castelnuovo-Mumford regularity of a permutation. St001438The number of missing boxes of a skew partition. St001569The maximal modular displacement of a permutation. St001706The number of closed sets in a graph. St001434The number of negative sum pairs of a signed permutation. St001621The number of atoms of a lattice. St000973The length of the boundary of an ordered tree. St001249Sum of the odd parts of a partition. St001769The reflection length of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000302The determinant of the distance matrix of a connected graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001866The nesting alignments of a signed permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001834The number of non-isomorphic minors of a graph. St001926Sparre Andersen's position of the maximum of a signed permutation. St001673The degree of asymmetry of an integer composition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000522The number of 1-protected nodes of a rooted tree. St001409The maximal entry of a semistandard tableau. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001670The connected partition number of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001894The depth of a signed permutation. St000017The number of inversions of a standard tableau. St000035The number of left outer peaks of a permutation. St000387The matching number of a graph. St000521The number of distinct subtrees of an ordered tree. St000820The number of compositions obtained by rotating the composition. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000910The number of maximal chains of minimal length in a poset. St000918The 2-limited packing number of a graph. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001510The number of self-evacuating linear extensions of a finite poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001674The number of vertices of the largest induced star graph in the graph. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001863The number of weak excedances of a signed permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between e_i J and e_j J (the radical of the indecomposable projective modules). St001902The number of potential covers of a poset. St000168The number of internal nodes of an ordered tree. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001111The weak 2-dynamic chromatic number of a graph. St001393The induced matching number of a graph. St001423The number of distinct cubes in a binary word. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001716The 1-improper chromatic number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001903The number of fixed points of a parking function. 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. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001946The number of descents in a parking function. St000455The second largest eigenvalue of a graph if it is integral. St001115The number of even descents of a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001396Number of triples of incomparable elements in a finite poset. St001743The discrepancy of a graph. St001895The oddness of a signed permutation. St000274The number of perfect matchings of a graph. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001424The number of distinct squares in a binary word. St001734The lettericity of a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001964The interval resolution global dimension of a poset. St000260The radius of a connected graph. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by 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. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001877Number of indecomposable injective modules with projective dimension 2. St001488The number of corners of a skew partition. St000075The orbit size of a standard tableau under promotion. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000072The number of circled entries. St000077The number of boxed and circled entries. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001487The number of inner corners of a skew partition. St001642The Prague dimension of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000782The indicator function of whether a given perfect matching is an L & P matching. St000391The sum of the positions of the ones in a binary word. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St000845The maximal number of elements covered by an element in a poset. St000422The energy of a graph, if it is integral. St000537The cutwidth of a graph. St000568The hook number of a binary tree. St000846The maximal number of elements covering an element of a poset. St000919The number of maximal left branches of a binary tree. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000307The number of rowmotion orbits of a poset. St000442The maximal area to the right of an up step of a Dyck path. St000536The pathwidth of a graph. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001618The cardinality of the Frattini sublattice of a lattice. St001962The proper pathwidth of a graph. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001331The size of the minimal feedback vertex set. St001638The book thickness of a graph. St001669The number of single rises in a Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000361The second Zagreb index of a graph.
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