Your data matches 141 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000734
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000503
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1],[2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1,2,3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1,3],[2]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2],[3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
Description
The maximal difference between two elements in a common block.
Matching statistic: St000728
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000728: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1],[2]]
 => [[1,2]]
 => {{1,2}}
 => 1
[[1,2,3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1,3],[2]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2],[3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1
Description
The dimension of a set partition. This is the sum of the lengths of the arcs of a set partition. Equivalently, one obtains that this is the sum of the maximal entries of the blocks minus the sum of the minimal entries of the blocks. A slightly shifted definition of the dimension is [[St000572]].
Matching statistic: St001415
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St001415: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => 0 => 1
[[1],[2]]
 => [[1,2]]
 => 0 => 1
[[1,2,3]]
 => [[1,2,3]]
 => 00 => 2
[[1,3],[2]]
 => [[1,2],[3]]
 => 01 => 1
[[1,2],[3]]
 => [[1,2,3]]
 => 00 => 2
[[1],[2],[3]]
 => [[1,2],[3]]
 => 01 => 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => 000 => 3
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => 010 => 3
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => 001 => 2
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => 000 => 3
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => 010 => 3
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => 000 => 3
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 011 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 010 => 3
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => 001 => 2
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => 011 => 1
Description
The length of the longest palindromic prefix of a binary word.
Matching statistic: St000178
(load all 2 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000178: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => [[2,0],[1]]
 => 0 = 1 - 1
[[1],[2]]
 => [[1,2]]
 => [[2,0],[1]]
 => 0 = 1 - 1
[[1,2,3]]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 1 = 2 - 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 0 = 1 - 1
[[1,2],[3]]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 1 = 2 - 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 0 = 1 - 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => 2 = 3 - 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 1 = 2 - 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => 2 = 3 - 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => 2 = 3 - 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 0 = 1 - 1
Description
Number of free entries. The ''tiling'' of a pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called ''tiles'', and each entry in a pattern belong to exactly one tile. A tile is ''free'' if it do not intersect any of the first and the last row. This statistic is the total number of entries that belong to a free tile.
Matching statistic: St000505
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => {{1,2}}
 => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => {{1,2}}
 => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => {{1,2,3}}
 => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => {{1,2},{3}}
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 2 = 1 + 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000724
(load all 4 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000724: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => [1,2] => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2 = 1 + 1
Description
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. Associate an increasing binary tree to the permutation using [[Mp00061]]. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1]. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum ([[St000060]]), see also [3].
Matching statistic: St000727
(load all 3 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => [1,2] => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2 = 1 + 1
Description
The largest label of a leaf in the binary search tree associated with the permutation. Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
Matching statistic: St000738
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => [[1],[2]]
 => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => [[1],[2]]
 => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000740
(load all 3 compositions to match this statistic)
Mapping
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => [[1,2]]
 => [1,2] => 2 = 1 + 1
[[1],[2]]
 => [[1,2]]
 => [1,2] => 2 = 1 + 1
[[1,2,3]]
 => [[1,2,3]]
 => [1,2,3] => 3 = 2 + 1
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1,2],[3]]
 => [[1,2,3]]
 => [1,2,3] => 3 = 2 + 1
[[1],[2],[3]]
 => [[1,2],[3]]
 => [3,1,2] => 2 = 1 + 1
[[1,2,3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,3,4],[2]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2,4],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1,2,3],[4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,3],[2,4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2],[3,4]]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 3 + 1
[[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2 = 1 + 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => 4 = 3 + 1
[[1,2],[3],[4]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3 = 2 + 1
[[1],[2],[3],[4]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2 = 1 + 1
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000060The greater neighbor of the maximum. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000246The number of non-inversions of a permutation. St000293The number of inversions of a binary word. St000316The number of non-left-to-right-maxima of a permutation. St000327The number of cover relations in a poset. St000339The maf index of a permutation. St000626The minimal period of a binary word. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000653The last descent of a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001246The maximal difference between two consecutive entries of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001726The number of visible inversions of a permutation. St000054The first entry of the permutation. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000235The number of indices that are not cyclical small weak excedances. St000338The number of pixed points of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000463The number of admissible inversions of a permutation. St000471The sum of the ascent tops of a permutation. St000501The size of the first part in the decomposition of a permutation. St000507The number of ascents of a standard tableau. St000708The product of the parts of an integer partition. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000736The last entry in the first row of a semistandard tableau. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000839The largest opener of a set partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000921The number of internal inversions of a binary word. St001082The number of boxed occurrences of 123 in a permutation. St001279The sum of the parts of an integer partition that are at least two. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001497The position of the largest weak excedence of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001727The number of invisible inversions of a permutation. St001959The product of the heights of the peaks of a Dyck path. St000176The total number of tiles in the Gelfand-Tsetlin pattern. 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. St001623The number of doubly irreducible elements of a lattice. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000806The semiperimeter of the associated bargraph. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000478Another 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000699The toughness times the least common multiple of 1,. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. 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. St001545The second Elser number of a connected graph. St001651The Frankl number of a lattice. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000466The Gutman (or modified Schultz) index of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000464The Schultz index of a connected graph. St000467The hyper-Wiener index of a connected graph.