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Your data matches 828 different statistics following compositions of up to 3 maps.
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Matching statistic: St000291
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St000291: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000291: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0 => 0 = -1 + 1
[[1],[2]]
=> 1 => 0 = -1 + 1
[[1,2,3]]
=> 00 => 0 = -1 + 1
[[1,3],[2]]
=> 10 => 1 = 0 + 1
[[1,2],[3]]
=> 01 => 0 = -1 + 1
[[1],[2],[3]]
=> 11 => 0 = -1 + 1
[[1,2,3,4]]
=> 000 => 0 = -1 + 1
[[1,2,4],[3]]
=> 010 => 1 = 0 + 1
[[1,2,3],[4]]
=> 001 => 0 = -1 + 1
[[1,2],[3,4]]
=> 010 => 1 = 0 + 1
[[1,4],[2],[3]]
=> 110 => 1 = 0 + 1
[[1,2],[3],[4]]
=> 011 => 0 = -1 + 1
[[1],[2],[3],[4]]
=> 111 => 0 = -1 + 1
[[1,2,3,4,5]]
=> 0000 => 0 = -1 + 1
[[1,2,3,5],[4]]
=> 0010 => 1 = 0 + 1
[[1,2,3,4],[5]]
=> 0001 => 0 = -1 + 1
[[1,2,3],[4,5]]
=> 0010 => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> 0110 => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> 0011 => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> 0110 => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> 1110 => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> 0111 => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> 00000 => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> 00001 => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> 00010 => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> 00110 => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> 00011 => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> 00110 => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> 01110 => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> 00111 => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> 01110 => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> 11110 => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> 01111 => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> 11111 => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> 000000 => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> 000010 => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> 000001 => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> 000010 => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> 000110 => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> 000011 => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> 000110 => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> 001110 => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> 000111 => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> 001110 => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> 011110 => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> 001111 => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> 011110 => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> 111110 => 1 = 0 + 1
Description
The number of descents of a binary word.
Matching statistic: St000766
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St000766: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000766: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 0 = -1 + 1
[[1],[2]]
=> [2] => 0 = -1 + 1
[[1,2,3]]
=> [3] => 0 = -1 + 1
[[1,3],[2]]
=> [2,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3] => 0 = -1 + 1
[[1],[2],[3]]
=> [3] => 0 = -1 + 1
[[1,2,3,4]]
=> [4] => 0 = -1 + 1
[[1,2,4],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4] => 0 = -1 + 1
[[1,2],[3,4]]
=> [3,1] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1],[2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1,2,3,4,5]]
=> [5] => 0 = -1 + 1
[[1,2,3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3],[4,5]]
=> [4,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
Description
The number of inversions of an integer composition.
This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.
Matching statistic: St000769
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St000769: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000769: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 0 = -1 + 1
[[1],[2]]
=> [2] => 0 = -1 + 1
[[1,2,3]]
=> [3] => 0 = -1 + 1
[[1,3],[2]]
=> [2,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3] => 0 = -1 + 1
[[1],[2],[3]]
=> [3] => 0 = -1 + 1
[[1,2,3,4]]
=> [4] => 0 = -1 + 1
[[1,2,4],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4] => 0 = -1 + 1
[[1,2],[3,4]]
=> [3,1] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1],[2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1,2,3,4,5]]
=> [5] => 0 = -1 + 1
[[1,2,3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3],[4,5]]
=> [4,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
Description
The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see [[St000008]].
Matching statistic: St000875
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St000875: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000875: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0 => 0 = -1 + 1
[[1],[2]]
=> 1 => 0 = -1 + 1
[[1,2,3]]
=> 00 => 0 = -1 + 1
[[1,3],[2]]
=> 10 => 1 = 0 + 1
[[1,2],[3]]
=> 01 => 0 = -1 + 1
[[1],[2],[3]]
=> 11 => 0 = -1 + 1
[[1,2,3,4]]
=> 000 => 0 = -1 + 1
[[1,2,4],[3]]
=> 010 => 1 = 0 + 1
[[1,2,3],[4]]
=> 001 => 0 = -1 + 1
[[1,2],[3,4]]
=> 010 => 1 = 0 + 1
[[1,4],[2],[3]]
=> 110 => 1 = 0 + 1
[[1,2],[3],[4]]
=> 011 => 0 = -1 + 1
[[1],[2],[3],[4]]
=> 111 => 0 = -1 + 1
[[1,2,3,4,5]]
=> 0000 => 0 = -1 + 1
[[1,2,3,5],[4]]
=> 0010 => 1 = 0 + 1
[[1,2,3,4],[5]]
=> 0001 => 0 = -1 + 1
[[1,2,3],[4,5]]
=> 0010 => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> 0110 => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> 0011 => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> 0110 => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> 1110 => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> 0111 => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> 00000 => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> 00001 => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> 00010 => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> 00110 => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> 00011 => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> 00110 => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> 01110 => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> 00111 => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> 01110 => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> 11110 => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> 01111 => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> 11111 => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> 000000 => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> 000010 => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> 000001 => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> 000010 => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> 000110 => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> 000011 => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> 000110 => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> 001110 => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> 000111 => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> 001110 => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> 011110 => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> 001111 => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> 011110 => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> 111110 => 1 = 0 + 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001421
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St001421: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001421: Binary words ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0 => 0 = -1 + 1
[[1],[2]]
=> 1 => 0 = -1 + 1
[[1,2,3]]
=> 00 => 0 = -1 + 1
[[1,3],[2]]
=> 10 => 1 = 0 + 1
[[1,2],[3]]
=> 01 => 0 = -1 + 1
[[1],[2],[3]]
=> 11 => 0 = -1 + 1
[[1,2,3,4]]
=> 000 => 0 = -1 + 1
[[1,2,4],[3]]
=> 010 => 1 = 0 + 1
[[1,2,3],[4]]
=> 001 => 0 = -1 + 1
[[1,2],[3,4]]
=> 010 => 1 = 0 + 1
[[1,4],[2],[3]]
=> 110 => 1 = 0 + 1
[[1,2],[3],[4]]
=> 011 => 0 = -1 + 1
[[1],[2],[3],[4]]
=> 111 => 0 = -1 + 1
[[1,2,3,4,5]]
=> 0000 => 0 = -1 + 1
[[1,2,3,5],[4]]
=> 0010 => 1 = 0 + 1
[[1,2,3,4],[5]]
=> 0001 => 0 = -1 + 1
[[1,2,3],[4,5]]
=> 0010 => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> 0110 => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> 0011 => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> 0110 => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> 1110 => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> 0111 => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> 1111 => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> 00000 => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> 00010 => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> 00001 => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> 00010 => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> 00110 => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> 00011 => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> 00110 => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> 01110 => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> 00111 => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> 01110 => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> 11110 => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> 01111 => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> 11111 => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> 000000 => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> 000010 => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> 000001 => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> 000010 => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> 000110 => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> 000011 => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> 000110 => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> 001110 => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> 000111 => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> 001110 => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> 011110 => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> 001111 => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> 011110 => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> 111110 => 1 = 0 + 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001673
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St001673: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001673: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 0 = -1 + 1
[[1],[2]]
=> [2] => 0 = -1 + 1
[[1,2,3]]
=> [3] => 0 = -1 + 1
[[1,3],[2]]
=> [2,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3] => 0 = -1 + 1
[[1],[2],[3]]
=> [3] => 0 = -1 + 1
[[1,2,3,4]]
=> [4] => 0 = -1 + 1
[[1,2,4],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4] => 0 = -1 + 1
[[1,2],[3,4]]
=> [3,1] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1],[2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1,2,3,4,5]]
=> [5] => 0 = -1 + 1
[[1,2,3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3],[4,5]]
=> [4,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
Description
The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St001777
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St001777: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001777: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 0 = -1 + 1
[[1],[2]]
=> [2] => 0 = -1 + 1
[[1,2,3]]
=> [3] => 0 = -1 + 1
[[1,3],[2]]
=> [2,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3] => 0 = -1 + 1
[[1],[2],[3]]
=> [3] => 0 = -1 + 1
[[1,2,3,4]]
=> [4] => 0 = -1 + 1
[[1,2,4],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4] => 0 = -1 + 1
[[1,2],[3,4]]
=> [3,1] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1],[2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1,2,3,4,5]]
=> [5] => 0 = -1 + 1
[[1,2,3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3],[4,5]]
=> [4,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
Description
The number of weak descents in an integer composition.
A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.
Matching statistic: St001931
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St001931: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001931: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 0 = -1 + 1
[[1],[2]]
=> [2] => 0 = -1 + 1
[[1,2,3]]
=> [3] => 0 = -1 + 1
[[1,3],[2]]
=> [2,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3] => 0 = -1 + 1
[[1],[2],[3]]
=> [3] => 0 = -1 + 1
[[1,2,3,4]]
=> [4] => 0 = -1 + 1
[[1,2,4],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4] => 0 = -1 + 1
[[1,2],[3,4]]
=> [3,1] => 1 = 0 + 1
[[1,4],[2],[3]]
=> [3,1] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1],[2],[3],[4]]
=> [4] => 0 = -1 + 1
[[1,2,3,4,5]]
=> [5] => 0 = -1 + 1
[[1,2,3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3],[4,5]]
=> [4,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2],[3,5],[4]]
=> [4,1] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
=> [4,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1],[2],[3],[4],[5]]
=> [5] => 0 = -1 + 1
[[1,2,3,4,5,6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4],[5,6]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3],[4,6],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => 1 = 0 + 1
[[1,2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6] => 0 = -1 + 1
[[1,2,3,4,5,6,7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5,6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4,5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3,4],[5,7],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,2,3],[4],[5],[6],[7]]
=> [7] => 0 = -1 + 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => 1 = 0 + 1
Description
The weak major index of an integer composition regarded as a word.
This is the sum of the positions of the weak descents, regarding the composition as a word. That is, for a composition $c = (c_1,\dots,c_n)$,
$$
\sum_{\substack{1\leq i < n\\ c_i\geq c_{i+1}}} i.
$$
Matching statistic: St000758
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00207: Standard tableaux āhorizontal strip sizesā¶ Integer compositions
St000758: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000758: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 1 = -1 + 2
[[1],[2]]
=> [1,1] => 1 = -1 + 2
[[1,2,3]]
=> [3] => 1 = -1 + 2
[[1,3],[2]]
=> [1,2] => 2 = 0 + 2
[[1,2],[3]]
=> [2,1] => 1 = -1 + 2
[[1],[2],[3]]
=> [1,1,1] => 1 = -1 + 2
[[1,2,3,4]]
=> [4] => 1 = -1 + 2
[[1,2,4],[3]]
=> [2,2] => 2 = 0 + 2
[[1,2,3],[4]]
=> [3,1] => 1 = -1 + 2
[[1,2],[3,4]]
=> [2,2] => 2 = 0 + 2
[[1,4],[2],[3]]
=> [1,1,2] => 2 = 0 + 2
[[1,2],[3],[4]]
=> [2,1,1] => 1 = -1 + 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => 1 = -1 + 2
[[1,2,3,4,5]]
=> [5] => 1 = -1 + 2
[[1,2,3,5],[4]]
=> [3,2] => 2 = 0 + 2
[[1,2,3,4],[5]]
=> [4,1] => 1 = -1 + 2
[[1,2,3],[4,5]]
=> [3,2] => 2 = 0 + 2
[[1,2,5],[3],[4]]
=> [2,1,2] => 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [3,1,1] => 1 = -1 + 2
[[1,2],[3,5],[4]]
=> [2,1,2] => 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => 2 = 0 + 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1 = -1 + 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1 = -1 + 2
[[1,2,3,4,5,6]]
=> [6] => 1 = -1 + 2
[[1,2,3,4,6],[5]]
=> [4,2] => 2 = 0 + 2
[[1,2,3,4,5],[6]]
=> [5,1] => 1 = -1 + 2
[[1,2,3,4],[5,6]]
=> [4,2] => 2 = 0 + 2
[[1,2,3,6],[4],[5]]
=> [3,1,2] => 2 = 0 + 2
[[1,2,3,4],[5],[6]]
=> [4,1,1] => 1 = -1 + 2
[[1,2,3],[4,6],[5]]
=> [3,1,2] => 2 = 0 + 2
[[1,2,6],[3],[4],[5]]
=> [2,1,1,2] => 2 = 0 + 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1 = -1 + 2
[[1,2],[3,6],[4],[5]]
=> [2,1,1,2] => 2 = 0 + 2
[[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 2 = 0 + 2
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1 = -1 + 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1 = -1 + 2
[[1,2,3,4,5,6,7]]
=> [7] => 1 = -1 + 2
[[1,2,3,4,5,7],[6]]
=> [5,2] => 2 = 0 + 2
[[1,2,3,4,5,6],[7]]
=> [6,1] => 1 = -1 + 2
[[1,2,3,4,5],[6,7]]
=> [5,2] => 2 = 0 + 2
[[1,2,3,4,7],[5],[6]]
=> [4,1,2] => 2 = 0 + 2
[[1,2,3,4,5],[6],[7]]
=> [5,1,1] => 1 = -1 + 2
[[1,2,3,4],[5,7],[6]]
=> [4,1,2] => 2 = 0 + 2
[[1,2,3,7],[4],[5],[6]]
=> [3,1,1,2] => 2 = 0 + 2
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 1 = -1 + 2
[[1,2,3],[4,7],[5],[6]]
=> [3,1,1,2] => 2 = 0 + 2
[[1,2,7],[3],[4],[5],[6]]
=> [2,1,1,1,2] => 2 = 0 + 2
[[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1 = -1 + 2
[[1,2],[3,7],[4],[5],[6]]
=> [2,1,1,1,2] => 2 = 0 + 2
[[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => 2 = 0 + 2
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000760
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00295: Standard tableaux āvalley compositionā¶ Integer compositions
St000760: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000760: Integer compositions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [2] => 1 = -1 + 2
[[1],[2]]
=> [2] => 1 = -1 + 2
[[1,2,3]]
=> [3] => 1 = -1 + 2
[[1,3],[2]]
=> [2,1] => 2 = 0 + 2
[[1,2],[3]]
=> [3] => 1 = -1 + 2
[[1],[2],[3]]
=> [3] => 1 = -1 + 2
[[1,2,3,4]]
=> [4] => 1 = -1 + 2
[[1,2,4],[3]]
=> [3,1] => 2 = 0 + 2
[[1,2,3],[4]]
=> [4] => 1 = -1 + 2
[[1,2],[3,4]]
=> [3,1] => 2 = 0 + 2
[[1,4],[2],[3]]
=> [3,1] => 2 = 0 + 2
[[1,2],[3],[4]]
=> [4] => 1 = -1 + 2
[[1],[2],[3],[4]]
=> [4] => 1 = -1 + 2
[[1,2,3,4,5]]
=> [5] => 1 = -1 + 2
[[1,2,3,5],[4]]
=> [4,1] => 2 = 0 + 2
[[1,2,3,4],[5]]
=> [5] => 1 = -1 + 2
[[1,2,3],[4,5]]
=> [4,1] => 2 = 0 + 2
[[1,2,5],[3],[4]]
=> [4,1] => 2 = 0 + 2
[[1,2,3],[4],[5]]
=> [5] => 1 = -1 + 2
[[1,2],[3,5],[4]]
=> [4,1] => 2 = 0 + 2
[[1,5],[2],[3],[4]]
=> [4,1] => 2 = 0 + 2
[[1,2],[3],[4],[5]]
=> [5] => 1 = -1 + 2
[[1],[2],[3],[4],[5]]
=> [5] => 1 = -1 + 2
[[1,2,3,4,5,6]]
=> [6] => 1 = -1 + 2
[[1,2,3,4,6],[5]]
=> [5,1] => 2 = 0 + 2
[[1,2,3,4,5],[6]]
=> [6] => 1 = -1 + 2
[[1,2,3,4],[5,6]]
=> [5,1] => 2 = 0 + 2
[[1,2,3,6],[4],[5]]
=> [5,1] => 2 = 0 + 2
[[1,2,3,4],[5],[6]]
=> [6] => 1 = -1 + 2
[[1,2,3],[4,6],[5]]
=> [5,1] => 2 = 0 + 2
[[1,2,6],[3],[4],[5]]
=> [5,1] => 2 = 0 + 2
[[1,2,3],[4],[5],[6]]
=> [6] => 1 = -1 + 2
[[1,2],[3,6],[4],[5]]
=> [5,1] => 2 = 0 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,1] => 2 = 0 + 2
[[1,2],[3],[4],[5],[6]]
=> [6] => 1 = -1 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6] => 1 = -1 + 2
[[1,2,3,4,5,6,7]]
=> [7] => 1 = -1 + 2
[[1,2,3,4,5,7],[6]]
=> [6,1] => 2 = 0 + 2
[[1,2,3,4,5,6],[7]]
=> [7] => 1 = -1 + 2
[[1,2,3,4,5],[6,7]]
=> [6,1] => 2 = 0 + 2
[[1,2,3,4,7],[5],[6]]
=> [6,1] => 2 = 0 + 2
[[1,2,3,4,5],[6],[7]]
=> [7] => 1 = -1 + 2
[[1,2,3,4],[5,7],[6]]
=> [6,1] => 2 = 0 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,1] => 2 = 0 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7] => 1 = -1 + 2
[[1,2,3],[4,7],[5],[6]]
=> [6,1] => 2 = 0 + 2
[[1,2,7],[3],[4],[5],[6]]
=> [6,1] => 2 = 0 + 2
[[1,2,3],[4],[5],[6],[7]]
=> [7] => 1 = -1 + 2
[[1,2],[3,7],[4],[5],[6]]
=> [6,1] => 2 = 0 + 2
[[1,7],[2],[3],[4],[5],[6]]
=> [6,1] => 2 = 0 + 2
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition.
By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
The following 818 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000767The number of runs in an integer composition. St000820The number of compositions obtained by rotating the composition. St000903The number of different parts of an integer composition. St001486The number of corners of the ribbon associated with an integer composition. St000008The major index of the composition. St000053The number of valleys of the Dyck path. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000552The number of cut vertices of a graph. St000674The number of hills of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000761The number of ascents in an integer composition. St000948The chromatic discriminant of a graph. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners 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. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001214The aft of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001459The number of zero columns in the nullspace of a graph. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001592The maximal number of simple paths between any two different vertices of a graph. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001712The number of natural descents of a standard Young tableau. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000288The number of ones in a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000346The number of coarsenings of a partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000691The number of changes of a binary word. St000701The protection number of a binary tree. St000722The number of different neighbourhoods in a graph. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000808The number of up steps of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. 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. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to 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 Dyck path as follows:
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001316The domatic number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001432The order dimension of the partition. St001471The magnitude of a Dyck path. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001675The number of parts equal to the part in the reversed composition. St001716The 1-improper chromatic number of a graph. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000981The length of the longest zigzag subpath. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000455The second largest eigenvalue of a graph if it is integral. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000081The number of edges of a graph. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000183The side length of the Durfee square of an integer partition. St000204The number of internal nodes of a binary tree. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000225Difference between largest and smallest parts in a partition. St000228The size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000274The number of perfect matchings of a graph. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000310The minimal degree of a vertex of a graph. St000313The number of degree 2 vertices of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000361The second Zagreb index of a graph. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000378The diagonal inversion number of an integer partition. 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. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000454The largest eigenvalue of a graph if it is integral. St000459The hook length of the base cell of a partition. St000466The Gutman (or modified Schultz) index of a connected graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000497The lcb statistic of a set partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. 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. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000628The balance of a binary word. St000632The jump number of the poset. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000682The Grundy value of Welter's game on a binary word. 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. St000693The modular (standard) major index of a standard tableau. St000741The Colin de VerdiĆØre graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000768The number of peaks in an integer composition. St000778The metric dimension of a graph. St000784The maximum of the length and the largest part of the integer partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000874The position of the last double rise in a Dyck path. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000940The number of characters of the symmetric group whose value on the partition is zero. St000947The major index east count of a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000992The alternating sum of the parts of an integer partition. 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. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001071The beta invariant of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001127The sum of the squares of the parts of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001161The major index north count of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001280The number of parts of an integer partition that are at least two. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001350Half of the Albertson index of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001424The number of distinct squares in a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001479The number of bridges of a graph. St001485The modular major index of a binary word. St001512The minimum rank of a graph. St001524The degree of symmetry of a binary word. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001649The length of a longest trail in a graph. St001657The number of twos in an integer partition. St001691The number of kings in a graph. St001695The natural comajor index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. 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. St001721The degree of a binary word. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001781The interlacing number of a set partition. St001783The number of odd automorphisms of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the TurĆ”n graph. St001799The number of proper separations of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001869The maximum cut size of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000048The multinomial of the parts of a partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000086The number of subgraphs. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000108The number of partitions contained in the given partition. St000146The Andrews-Garvan crank of a partition. St000179The product of the hook lengths of the integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000273The domination number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000296The length of the symmetric border of a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000343The number of spanning subgraphs of a graph. St000345The number of refinements of a partition. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000413The number of ordered trees with the same underlying unordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000522The number of 1-protected nodes of a rooted tree. St000527The width of the poset. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000544The cop number of a graph. St000553The number of blocks of a graph. St000568The hook number of a binary tree. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000636The hull number of a graph. St000657The smallest part of an integer composition. St000668The least common multiple of the parts of the partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000753The Grundy value for the game of Kayles on a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. 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. St000905The number of different multiplicities of parts of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000920The logarithmic height of a Dyck path. St000922The minimal number such that all substrings of this length are unique. St000926The clique-coclique number of a graph. St000933The number of multipartitions of sizes given by an integer partition. St000935The number of ordered refinements of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000972The composition number of a graph. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St000993The multiplicity of the largest part of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001128The exponens consonantiae of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. 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. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001360The number of covering relations in Young's lattice below a partition. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001366The maximal multiplicity of a degree of a vertex of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001368The number of vertices of maximal degree in a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001437The flex of a binary word. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. 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. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001725The harmonious chromatic number of a graph. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001802The number of endomorphisms of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St000294The number of distinct factors of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000391The sum of the positions of the ones in a binary word. St000439The position of the first down step of a Dyck path. St000518The number of distinct subsequences in a binary word. St000792The Grundy value for the game of ruler on a binary word. St000969We 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 $c_{n-1}$. 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. St001658The total number of rook placements on a Ferrers board. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. 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. St001463The number of distinct columns in the nullspace of a graph. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000371The number of mid points of decreasing 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 $3$. St000463The number of admissible inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000287The number of connected components of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001734The lettericity of a graph. St000647The number of big descents of a permutation. St001394The genus of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000779The tier of a permutation. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000486The number of cycles of length at least 3 of a permutation. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the 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. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000675The number of centered multitunnels of a Dyck path. St001096The size of the overlap set of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000670The reversal length of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. 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. St000845The maximal number of elements covered by an element in a poset. St000354The number of recoils of a permutation. St000446The disorder of a permutation. St000462The major index minus the number of excedences of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of 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. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St000961The shifted major index of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000801The number of occurrences of the vincular pattern |312 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. St000096The number of spanning trees of a graph. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000523The number of 2-protected nodes of a rooted tree. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001742The difference of the maximal and the minimal degree in a graph. St001859The number of factors of the Stanley symmetric function associated with a 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). St001874Lusztig's a-function for the symmetric group. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000700The protection number of an ordered tree. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001642The Prague dimension of a graph. St001917The order of toric promotion on the set of labellings of a graph. St001959The product of the heights of the peaks of a Dyck path. St000397The Strahler number of a rooted tree. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000117The number of centered tunnels of a Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension 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. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000422The energy of a graph, if it is integral. St000456The monochromatic index of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph. St000145The Dyson rank 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. St000379The number of Hamiltonian cycles in a graph. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000699The toughness times the least common multiple of 1,. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000934The 2-degree 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. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000284The Plancherel distribution on integer partitions. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001118The acyclic chromatic index of a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001943The sum of the squares of the hook lengths of an integer partition. St000464The Schultz index of a connected graph. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St000997The even-odd crank of an integer partition. 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. St001545The second Elser number of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001487The number of inner corners of a skew partition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001651The Frankl number of a lattice. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed 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 $eA$. 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 $eA$. St001569The maximal modular displacement of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001857The number of edges in the reduced word graph of a signed permutation. St001623The number of doubly irreducible elements of a lattice. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
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