Your data matches 395 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 8 compositions to match this statistic)
Mapping
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 2
[3]
 => 1
[2,1]
 => 2
[1,1,1]
 => 3
[4]
 => 1
[3,1]
 => 2
[2,2]
 => 2
[2,1,1]
 => 3
[1,1,1,1]
 => 4
[5]
 => 1
[4,1]
 => 2
[3,2]
 => 2
[3,1,1]
 => 3
[2,2,1]
 => 3
[2,1,1,1]
 => 4
[1,1,1,1,1]
 => 5
[6]
 => 1
[5,1]
 => 2
[4,2]
 => 2
[4,1,1]
 => 3
[3,3]
 => 2
[3,2,1]
 => 3
[3,1,1,1]
 => 4
[2,2,2]
 => 3
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 5
[1,1,1,1,1,1]
 => 6
[7]
 => 1
[6,1]
 => 2
[5,2]
 => 2
[5,1,1]
 => 3
[4,3]
 => 2
[4,2,1]
 => 3
[4,1,1,1]
 => 4
[3,3,1]
 => 3
[3,2,2]
 => 3
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 5
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 6
[1,1,1,1,1,1,1]
 => 7
[8]
 => 1
[7,1]
 => 2
[6,2]
 => 2
[6,1,1]
 => 3
[5,3]
 => 2
[5,2,1]
 => 3
Description
The length of the partition.
Matching statistic: St000147
(load all 8 compositions to match this statistic)
Mapping
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 1
[4]
 => 4
[3,1]
 => 3
[2,2]
 => 2
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 5
[4,1]
 => 4
[3,2]
 => 3
[3,1,1]
 => 3
[2,2,1]
 => 2
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 6
[5,1]
 => 5
[4,2]
 => 4
[4,1,1]
 => 4
[3,3]
 => 3
[3,2,1]
 => 3
[3,1,1,1]
 => 3
[2,2,2]
 => 2
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 7
[6,1]
 => 6
[5,2]
 => 5
[5,1,1]
 => 5
[4,3]
 => 4
[4,2,1]
 => 4
[4,1,1,1]
 => 4
[3,3,1]
 => 3
[3,2,2]
 => 3
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 3
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 8
[7,1]
 => 7
[6,2]
 => 6
[6,1,1]
 => 6
[5,3]
 => 5
[5,2,1]
 => 5
Description
The largest part of an integer partition.
Matching statistic: St000378
(load all 3 compositions to match this statistic)
Mapping
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 2
[2,1]
 => 3
[1,1,1]
 => 1
[4]
 => 2
[3,1]
 => 4
[2,2]
 => 3
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 2
[4,1]
 => 3
[3,2]
 => 5
[3,1,1]
 => 4
[2,2,1]
 => 3
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 2
[5,1]
 => 3
[4,2]
 => 5
[4,1,1]
 => 4
[3,3]
 => 4
[3,2,1]
 => 6
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 2
[6,1]
 => 3
[5,2]
 => 4
[5,1,1]
 => 3
[4,3]
 => 5
[4,2,1]
 => 7
[4,1,1,1]
 => 4
[3,3,1]
 => 6
[3,2,2]
 => 5
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 3
[2,2,2,1]
 => 3
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 2
[7,1]
 => 3
[6,2]
 => 4
[6,1,1]
 => 3
[5,3]
 => 5
[5,2,1]
 => 5
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000007
(load all 15 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1]
 => []
 => []
 => [] => 0 = 1 - 1
[2]
 => []
 => []
 => [] => 0 = 1 - 1
[1,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[3]
 => []
 => []
 => [] => 0 = 1 - 1
[2,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[1,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[4]
 => []
 => []
 => [] => 0 = 1 - 1
[3,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[2,2]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[2,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 4 - 1
[5]
 => []
 => []
 => [] => 0 = 1 - 1
[4,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[3,2]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[3,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 4 - 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 4 = 5 - 1
[6]
 => []
 => []
 => [] => 0 = 1 - 1
[5,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[4,2]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[4,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[3,3]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 2 - 1
[3,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 4 - 1
[2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3 = 4 - 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 4 = 5 - 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 5 = 6 - 1
[7]
 => []
 => []
 => [] => 0 = 1 - 1
[6,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[5,2]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[5,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[4,3]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 2 - 1
[4,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 4 - 1
[3,3,1]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2 = 3 - 1
[3,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2 = 3 - 1
[3,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 3 = 4 - 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 4 = 5 - 1
[2,2,2,1]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 3 = 4 - 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4 = 5 - 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 5 = 6 - 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 6 = 7 - 1
[8]
 => []
 => []
 => [] => 0 = 1 - 1
[7,1]
 => [1]
 => [[1]]
 => [1] => 1 = 2 - 1
[6,2]
 => [2]
 => [[1,2]]
 => [1,2] => 1 = 2 - 1
[6,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 3 - 1
[5,3]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 2 - 1
[5,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 3 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? => ? = 12 - 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000507
(load all 4 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1]
 => []
 => []
 => []
 => 0 = 1 - 1
[2]
 => []
 => []
 => []
 => 0 = 1 - 1
[1,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[3]
 => []
 => []
 => []
 => 0 = 1 - 1
[2,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[1,1,1]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2 = 3 - 1
[4]
 => []
 => []
 => []
 => 0 = 1 - 1
[3,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[2,2]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1 = 2 - 1
[2,1,1]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3 = 4 - 1
[5]
 => []
 => []
 => []
 => 0 = 1 - 1
[4,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[3,2]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1 = 2 - 1
[3,1,1]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2 = 3 - 1
[2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 3 - 1
[2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3 = 4 - 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 4 = 5 - 1
[6]
 => []
 => []
 => []
 => 0 = 1 - 1
[5,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[4,2]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1 = 2 - 1
[4,1,1]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2 = 3 - 1
[3,3]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1 = 2 - 1
[3,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 3 - 1
[3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3 = 4 - 1
[2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2 = 3 - 1
[2,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3 = 4 - 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 4 = 5 - 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 5 = 6 - 1
[7]
 => []
 => []
 => []
 => 0 = 1 - 1
[6,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[5,2]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1 = 2 - 1
[5,1,1]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2 = 3 - 1
[4,3]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1 = 2 - 1
[4,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 3 - 1
[4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 3 = 4 - 1
[3,3,1]
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2 = 3 - 1
[3,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2 = 3 - 1
[3,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3 = 4 - 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 4 = 5 - 1
[2,2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4 = 5 - 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 5 = 6 - 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 6 = 7 - 1
[8]
 => []
 => []
 => []
 => 0 = 1 - 1
[7,1]
 => [1]
 => [[1]]
 => [[1]]
 => 1 = 2 - 1
[6,2]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1 = 2 - 1
[6,1,1]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 2 = 3 - 1
[5,3]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 1 = 2 - 1
[5,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 3 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? = 12 - 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000228
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 99%●distinct values known / distinct values provided: 83%
Values
[1]
 => 1 => [1] => [1]
 => 1
[2]
 => 0 => [1] => [1]
 => 1
[1,1]
 => 11 => [2] => [2]
 => 2
[3]
 => 1 => [1] => [1]
 => 1
[2,1]
 => 01 => [1,1] => [1,1]
 => 2
[1,1,1]
 => 111 => [3] => [3]
 => 3
[4]
 => 0 => [1] => [1]
 => 1
[3,1]
 => 11 => [2] => [2]
 => 2
[2,2]
 => 00 => [2] => [2]
 => 2
[2,1,1]
 => 011 => [1,2] => [2,1]
 => 3
[1,1,1,1]
 => 1111 => [4] => [4]
 => 4
[5]
 => 1 => [1] => [1]
 => 1
[4,1]
 => 01 => [1,1] => [1,1]
 => 2
[3,2]
 => 10 => [1,1] => [1,1]
 => 2
[3,1,1]
 => 111 => [3] => [3]
 => 3
[2,2,1]
 => 001 => [2,1] => [2,1]
 => 3
[2,1,1,1]
 => 0111 => [1,3] => [3,1]
 => 4
[1,1,1,1,1]
 => 11111 => [5] => [5]
 => 5
[6]
 => 0 => [1] => [1]
 => 1
[5,1]
 => 11 => [2] => [2]
 => 2
[4,2]
 => 00 => [2] => [2]
 => 2
[4,1,1]
 => 011 => [1,2] => [2,1]
 => 3
[3,3]
 => 11 => [2] => [2]
 => 2
[3,2,1]
 => 101 => [1,1,1] => [1,1,1]
 => 3
[3,1,1,1]
 => 1111 => [4] => [4]
 => 4
[2,2,2]
 => 000 => [3] => [3]
 => 3
[2,2,1,1]
 => 0011 => [2,2] => [2,2]
 => 4
[2,1,1,1,1]
 => 01111 => [1,4] => [4,1]
 => 5
[1,1,1,1,1,1]
 => 111111 => [6] => [6]
 => 6
[7]
 => 1 => [1] => [1]
 => 1
[6,1]
 => 01 => [1,1] => [1,1]
 => 2
[5,2]
 => 10 => [1,1] => [1,1]
 => 2
[5,1,1]
 => 111 => [3] => [3]
 => 3
[4,3]
 => 01 => [1,1] => [1,1]
 => 2
[4,2,1]
 => 001 => [2,1] => [2,1]
 => 3
[4,1,1,1]
 => 0111 => [1,3] => [3,1]
 => 4
[3,3,1]
 => 111 => [3] => [3]
 => 3
[3,2,2]
 => 100 => [1,2] => [2,1]
 => 3
[3,2,1,1]
 => 1011 => [1,1,2] => [2,1,1]
 => 4
[3,1,1,1,1]
 => 11111 => [5] => [5]
 => 5
[2,2,2,1]
 => 0001 => [3,1] => [3,1]
 => 4
[2,2,1,1,1]
 => 00111 => [2,3] => [3,2]
 => 5
[2,1,1,1,1,1]
 => 011111 => [1,5] => [5,1]
 => 6
[1,1,1,1,1,1,1]
 => 1111111 => [7] => [7]
 => 7
[8]
 => 0 => [1] => [1]
 => 1
[7,1]
 => 11 => [2] => [2]
 => 2
[6,2]
 => 00 => [2] => [2]
 => 2
[6,1,1]
 => 011 => [1,2] => [2,1]
 => 3
[5,3]
 => 11 => [2] => [2]
 => 2
[5,2,1]
 => 101 => [1,1,1] => [1,1,1]
 => 3
[1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => [11] => [11]
 => ? = 11
[2,1,1,1,1,1,1,1,1,1,1]
 => 01111111111 => [1,10] => [10,1]
 => ? ∊ {11,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111 => [12] => [12]
 => ? ∊ {11,12}
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000876
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1]
 => 1 => 0 => 0 => 1
[2]
 => 0 => 1 => 1 => 1
[1,1]
 => 11 => 00 => 00 => 2
[3]
 => 1 => 0 => 0 => 1
[2,1]
 => 01 => 10 => 01 => 2
[1,1,1]
 => 111 => 000 => 000 => 3
[4]
 => 0 => 1 => 1 => 1
[3,1]
 => 11 => 00 => 00 => 2
[2,2]
 => 00 => 11 => 11 => 2
[2,1,1]
 => 011 => 100 => 001 => 3
[1,1,1,1]
 => 1111 => 0000 => 0000 => 4
[5]
 => 1 => 0 => 0 => 1
[4,1]
 => 01 => 10 => 01 => 2
[3,2]
 => 10 => 01 => 01 => 2
[3,1,1]
 => 111 => 000 => 000 => 3
[2,2,1]
 => 001 => 110 => 011 => 3
[2,1,1,1]
 => 0111 => 1000 => 0001 => 4
[1,1,1,1,1]
 => 11111 => 00000 => 00000 => 5
[6]
 => 0 => 1 => 1 => 1
[5,1]
 => 11 => 00 => 00 => 2
[4,2]
 => 00 => 11 => 11 => 2
[4,1,1]
 => 011 => 100 => 001 => 3
[3,3]
 => 11 => 00 => 00 => 2
[3,2,1]
 => 101 => 010 => 001 => 3
[3,1,1,1]
 => 1111 => 0000 => 0000 => 4
[2,2,2]
 => 000 => 111 => 111 => 3
[2,2,1,1]
 => 0011 => 1100 => 0011 => 4
[2,1,1,1,1]
 => 01111 => 10000 => 00001 => 5
[1,1,1,1,1,1]
 => 111111 => 000000 => 000000 => 6
[7]
 => 1 => 0 => 0 => 1
[6,1]
 => 01 => 10 => 01 => 2
[5,2]
 => 10 => 01 => 01 => 2
[5,1,1]
 => 111 => 000 => 000 => 3
[4,3]
 => 01 => 10 => 01 => 2
[4,2,1]
 => 001 => 110 => 011 => 3
[4,1,1,1]
 => 0111 => 1000 => 0001 => 4
[3,3,1]
 => 111 => 000 => 000 => 3
[3,2,2]
 => 100 => 011 => 011 => 3
[3,2,1,1]
 => 1011 => 0100 => 0001 => 4
[3,1,1,1,1]
 => 11111 => 00000 => 00000 => 5
[2,2,2,1]
 => 0001 => 1110 => 0111 => 4
[2,2,1,1,1]
 => 00111 => 11000 => 00011 => 5
[2,1,1,1,1,1]
 => 011111 => 100000 => 000001 => 6
[1,1,1,1,1,1,1]
 => 1111111 => 0000000 => 0000000 => 7
[8]
 => 0 => 1 => 1 => 1
[7,1]
 => 11 => 00 => 00 => 2
[6,2]
 => 00 => 11 => 11 => 2
[6,1,1]
 => 011 => 100 => 001 => 3
[5,3]
 => 11 => 00 => 00 => 2
[5,2,1]
 => 101 => 010 => 001 => 3
[1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => 00000000000 => 00000000000 => ? = 11
[2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => 1100000000 => ? => ? ∊ {10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111 => 000000000000 => 000000000000 => ? ∊ {10,12}
Description
The number of factors in the Catalan decomposition 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 number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000885
(load all 2 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1]
 => 1 => 0 => 0 => 1
[2]
 => 0 => 1 => 1 => 1
[1,1]
 => 11 => 00 => 00 => 2
[3]
 => 1 => 0 => 0 => 1
[2,1]
 => 01 => 10 => 01 => 2
[1,1,1]
 => 111 => 000 => 000 => 3
[4]
 => 0 => 1 => 1 => 1
[3,1]
 => 11 => 00 => 00 => 2
[2,2]
 => 00 => 11 => 11 => 2
[2,1,1]
 => 011 => 100 => 001 => 3
[1,1,1,1]
 => 1111 => 0000 => 0000 => 4
[5]
 => 1 => 0 => 0 => 1
[4,1]
 => 01 => 10 => 01 => 2
[3,2]
 => 10 => 01 => 01 => 2
[3,1,1]
 => 111 => 000 => 000 => 3
[2,2,1]
 => 001 => 110 => 011 => 3
[2,1,1,1]
 => 0111 => 1000 => 0001 => 4
[1,1,1,1,1]
 => 11111 => 00000 => 00000 => 5
[6]
 => 0 => 1 => 1 => 1
[5,1]
 => 11 => 00 => 00 => 2
[4,2]
 => 00 => 11 => 11 => 2
[4,1,1]
 => 011 => 100 => 001 => 3
[3,3]
 => 11 => 00 => 00 => 2
[3,2,1]
 => 101 => 010 => 001 => 3
[3,1,1,1]
 => 1111 => 0000 => 0000 => 4
[2,2,2]
 => 000 => 111 => 111 => 3
[2,2,1,1]
 => 0011 => 1100 => 0011 => 4
[2,1,1,1,1]
 => 01111 => 10000 => 00001 => 5
[1,1,1,1,1,1]
 => 111111 => 000000 => 000000 => 6
[7]
 => 1 => 0 => 0 => 1
[6,1]
 => 01 => 10 => 01 => 2
[5,2]
 => 10 => 01 => 01 => 2
[5,1,1]
 => 111 => 000 => 000 => 3
[4,3]
 => 01 => 10 => 01 => 2
[4,2,1]
 => 001 => 110 => 011 => 3
[4,1,1,1]
 => 0111 => 1000 => 0001 => 4
[3,3,1]
 => 111 => 000 => 000 => 3
[3,2,2]
 => 100 => 011 => 011 => 3
[3,2,1,1]
 => 1011 => 0100 => 0001 => 4
[3,1,1,1,1]
 => 11111 => 00000 => 00000 => 5
[2,2,2,1]
 => 0001 => 1110 => 0111 => 4
[2,2,1,1,1]
 => 00111 => 11000 => 00011 => 5
[2,1,1,1,1,1]
 => 011111 => 100000 => 000001 => 6
[1,1,1,1,1,1,1]
 => 1111111 => 0000000 => 0000000 => 7
[8]
 => 0 => 1 => 1 => 1
[7,1]
 => 11 => 00 => 00 => 2
[6,2]
 => 00 => 11 => 11 => 2
[6,1,1]
 => 011 => 100 => 001 => 3
[5,3]
 => 11 => 00 => 00 => 2
[5,2,1]
 => 101 => 010 => 001 => 3
[1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => 00000000000 => 00000000000 => ? = 11
[2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => 1100000000 => ? => ? ∊ {10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111 => 000000000000 => 000000000000 => ? ∊ {10,12}
Description
The number of critical steps in the Catalan decomposition 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 number of critical steps $\ell + m$ in the Catalan factorisation. The distribution of this statistic on words of length $n$ is $$ (n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}. $$
Matching statistic: St001437
(load all 3 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 98%●distinct values known / distinct values provided: 92%
Values
[1]
 => 1 => 1 => 1 => 1
[2]
 => 0 => 0 => 0 => 1
[1,1]
 => 11 => 11 => 11 => 2
[3]
 => 1 => 1 => 1 => 1
[2,1]
 => 01 => 01 => 10 => 2
[1,1,1]
 => 111 => 111 => 111 => 3
[4]
 => 0 => 0 => 0 => 1
[3,1]
 => 11 => 11 => 11 => 2
[2,2]
 => 00 => 00 => 00 => 2
[2,1,1]
 => 011 => 011 => 110 => 3
[1,1,1,1]
 => 1111 => 1111 => 1111 => 4
[5]
 => 1 => 1 => 1 => 1
[4,1]
 => 01 => 01 => 10 => 2
[3,2]
 => 10 => 01 => 10 => 2
[3,1,1]
 => 111 => 111 => 111 => 3
[2,2,1]
 => 001 => 001 => 100 => 3
[2,1,1,1]
 => 0111 => 0111 => 1110 => 4
[1,1,1,1,1]
 => 11111 => 11111 => 11111 => 5
[6]
 => 0 => 0 => 0 => 1
[5,1]
 => 11 => 11 => 11 => 2
[4,2]
 => 00 => 00 => 00 => 2
[4,1,1]
 => 011 => 011 => 110 => 3
[3,3]
 => 11 => 11 => 11 => 2
[3,2,1]
 => 101 => 011 => 110 => 3
[3,1,1,1]
 => 1111 => 1111 => 1111 => 4
[2,2,2]
 => 000 => 000 => 000 => 3
[2,2,1,1]
 => 0011 => 0011 => 1100 => 4
[2,1,1,1,1]
 => 01111 => 01111 => 11110 => 5
[1,1,1,1,1,1]
 => 111111 => 111111 => 111111 => 6
[7]
 => 1 => 1 => 1 => 1
[6,1]
 => 01 => 01 => 10 => 2
[5,2]
 => 10 => 01 => 10 => 2
[5,1,1]
 => 111 => 111 => 111 => 3
[4,3]
 => 01 => 01 => 10 => 2
[4,2,1]
 => 001 => 001 => 100 => 3
[4,1,1,1]
 => 0111 => 0111 => 1110 => 4
[3,3,1]
 => 111 => 111 => 111 => 3
[3,2,2]
 => 100 => 001 => 100 => 3
[3,2,1,1]
 => 1011 => 0111 => 1110 => 4
[3,1,1,1,1]
 => 11111 => 11111 => 11111 => 5
[2,2,2,1]
 => 0001 => 0001 => 1000 => 4
[2,2,1,1,1]
 => 00111 => 00111 => 11100 => 5
[2,1,1,1,1,1]
 => 011111 => 011111 => 111110 => 6
[1,1,1,1,1,1,1]
 => 1111111 => 1111111 => 1111111 => 7
[8]
 => 0 => 0 => 0 => 1
[7,1]
 => 11 => 11 => 11 => 2
[6,2]
 => 00 => 00 => 00 => 2
[6,1,1]
 => 011 => 011 => 110 => 3
[5,3]
 => 11 => 11 => 11 => 2
[5,2,1]
 => 101 => 011 => 110 => 3
[1,1,1,1,1,1,1,1,1,1]
 => 1111111111 => 1111111111 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => ? => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1]
 => 1111111111 => 1111111111 => 1111111111 => ? ∊ {10,10,12}
[2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? => ? => ? ∊ {10,10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => 111111111111 => 111111111111 => 111111111111 => ? ∊ {10,10,12}
Description
The flex of a binary word. This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St000676
(load all 29 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 96%●distinct values known / distinct values provided: 75%
Values
[1]
 => []
 => []
 => []
 => 0 = 1 - 1
[2]
 => []
 => []
 => []
 => 0 = 1 - 1
[1,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[3]
 => []
 => []
 => []
 => 0 = 1 - 1
[2,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[1,1,1]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[4]
 => []
 => []
 => []
 => 0 = 1 - 1
[3,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[2,2]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[5]
 => []
 => []
 => []
 => 0 = 1 - 1
[4,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[3,2]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[3,1,1]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[2,2,1]
 => [2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[6]
 => []
 => []
 => []
 => 0 = 1 - 1
[5,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[4,2]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[4,1,1]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[3,3]
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,1]
 => [2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[2,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[7]
 => []
 => []
 => []
 => 0 = 1 - 1
[6,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[5,2]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[5,1,1]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[4,3]
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[4,2,1]
 => [2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[3,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 5 - 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 7 - 1
[8]
 => []
 => []
 => []
 => 0 = 1 - 1
[7,1]
 => [1]
 => [1]
 => [1,0]
 => 1 = 2 - 1
[6,2]
 => [2]
 => [1,1]
 => [1,1,0,0]
 => 1 = 2 - 1
[6,1,1]
 => [1,1]
 => [2]
 => [1,0,1,0]
 => 2 = 3 - 1
[5,3]
 => [3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[5,2,1]
 => [2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10 - 1
[2,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {9,10,11} - 1
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {9,10,11} - 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {9,10,11} - 1
[3,3,1,1,1,1,1,1]
 => [3,1,1,1,1,1,1]
 => [7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
[3,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
[3,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
[2,2,2,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1]
 => [8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
[2,2,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1]
 => [9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [11]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {8,9,9,10,10,11,12} - 1
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
The following 385 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000288The number of ones in a binary word. St000013The height of a Dyck path. St000738The first entry in the last row of a standard tableau. St000734The last entry in the first row of a standard tableau. St000439The position of the first down step of a Dyck path. St000382The first part of an integer composition. St001777The number of weak descents in an integer composition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000381The largest part of an integer composition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000012The area of a Dyck path. St000808The number of up steps of the associated bargraph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000839The largest opener of a set partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000730The maximal arc length of a set partition. 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. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000653The last descent of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. 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. St000011The number of touch points (or returns) of a Dyck path. St001814The number of partitions interlacing the given partition. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000691The number of changes of a binary word. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St001372The length of a longest cyclic run of ones of a binary word. St001484The number of singletons of an integer partition. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001486The number of corners of the ribbon associated with an integer composition. St000161The sum of the sizes of the right subtrees of a binary tree. St000326The position of the first one in a binary word after appending a 1 at the end. St000651The maximal size of a rise in a permutation. St001733The number of weak left to right maxima of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001497The position of the largest weak excedence of a permutation. St000041The number of nestings of a perfect matching. St001397Number of pairs of incomparable elements in a finite poset. St001389The number of partitions of the same length below the given integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000019The cardinality of the support of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000392The length of the longest run of ones in a binary word. St000546The number of global descents of a permutation. St000957The number of Bruhat lower covers of a permutation. St001268The size of the largest ordinal summand in the poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000297The number of leading ones in a binary word. St000528The height of a poset. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000652The maximal difference between successive positions of a permutation. St000306The bounce count of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000080The rank of the poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000018The number of inversions of a permutation. St000246The number of non-inversions of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000445The number of rises of length 1 of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St000925The number of topologically connected components of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St000067The inversion number of the alternating sign matrix. St000451The length of the longest pattern of the form k 1 2. St000548The number of different non-empty partial sums of an integer partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000521The number of distinct subtrees of an ordered tree. St000006The dinv of a Dyck path. St000060The greater neighbor of the maximum. St000316The number of non-left-to-right-maxima of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St000005The bounce statistic of a Dyck path. St000051The size of the left subtree of a binary tree. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000167The number of leaves of an ordered tree. St000204The number of internal nodes of a binary tree. St000216The absolute length of a permutation. St000304The load of a permutation. St000356The number of occurrences of the pattern 13-2. St000809The reduced reflection length of the permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000028The number of stack-sorts needed to sort a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000133The "bounce" of a permutation. St000159The number of distinct parts of the integer partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000443The number of long tunnels of a Dyck path. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000069The number of maximal elements of a poset. St000692Babson and Steingrímsson's statistic of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000105The number of blocks in the set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000686The finitistic dominant dimension of a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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: St001674The number of vertices of the largest induced star graph in the graph. St000211The rank of the set partition. St000820The number of compositions obtained by rotating the composition. 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 $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000308The height of the tree associated to a permutation. St000982The length of the longest constant subword. St000031The number of cycles in the cycle decomposition of a permutation. St000527The width of the poset. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000052The number of valleys of a Dyck path not on the x-axis. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000912The number of maximal antichains in a poset. St000632The jump number of the poset. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000654The first descent of a permutation. St000446The disorder of a permutation. St000702The number of weak deficiencies of a permutation. St000833The comajor index of a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000160The multiplicity of the smallest part of a partition. St000667The greatest common divisor of the parts of the partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001461The number of topologically connected components of the chord diagram of a permutation. St000153The number of adjacent cycles of a permutation. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001726The number of visible inversions of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000292The number of ascents of a binary word. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000097The order of the largest clique of the graph. St000383The last part of an integer composition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001462The number of factors of a standard tableaux under concatenation. St000542The number of left-to-right-minima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000062The length of the longest increasing subsequence of the permutation. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000990The first ascent of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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 $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000004The major index of a permutation. St000021The number of descents of a permutation. St000094The depth of an ordered tree. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000305The inverse major index of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000794The mak of a permutation. St000989The number of final rises of a permutation. St001153The number of blocks with even minimum in a set partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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 $eA$. 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. 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001584The area statistic between a Dyck path and its bounce path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000061The number of nodes on the left branch of a binary tree. St001346The number of parking functions that give the same permutation. St001581The achromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000071The number of maximal chains in a poset. St000068The number of minimal elements in a poset. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000505The biggest entry in the block containing the 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001571The Cartan determinant of the integer partition. St000746The number of pairs with odd minimum in a perfect matching. St001645The pebbling number of a connected graph. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St001062The maximal size of a block of a set partition. St000823The number of unsplittable factors of the set partition. St000503The maximal difference between two elements in a common block. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000172The Grundy number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000916The packing number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001829The common independence number of a graph. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. 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. St000647The number of big descents of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001963The tree-depth of a graph. St000868The aid statistic in the sense of Shareshian-Wachs. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. 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. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001557The number of inversions of the second entry of a permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001343The dimension of the reduced incidence algebra of a poset. St000787The number of flips required to make a perfect matching noncrossing. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001427The number of descents of a signed permutation. St000908The length of the shortest maximal antichain in a poset. St001717The largest size of an interval in a poset. St000914The sum of the values of the Möbius function of a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000710The number of big deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St000089The absolute variation of a composition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001589The nesting number of a perfect matching. St000039The number of crossings of a permutation. St000144The pyramid weight of the Dyck path. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000719The number of alignments in a perfect matching. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. 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. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001684The reduced word complexity of a permutation. St001692The number of vertices with higher degree than the average degree in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001746The coalition number of a graph. St000501The size of the first part in the decomposition of a permutation. St001555The order of a signed permutation. St001962The proper pathwidth of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001948The number of augmented double ascents of a permutation. 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. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000485The length of the longest cycle of a permutation. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001812The biclique partition number of a graph. St000628The balance of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001644The dimension of a graph. St001875The number of simple modules with projective dimension at most 1. St001638The book thickness of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001488The number of corners of a skew partition. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. 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 $eA$.