Your data matches 320 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000377
Mapping
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 2
[4]
 => 2
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 2
[1,1,1,1]
 => 3
[5]
 => 3
[4,1]
 => 2
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 4
[6]
 => 4
[5,1]
 => 3
[4,2]
 => 1
[4,1,1]
 => 2
[3,3]
 => 2
[3,2,1]
 => 0
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 5
[7]
 => 5
[6,1]
 => 4
[5,2]
 => 3
[5,1,1]
 => 4
[4,3]
 => 2
[4,2,1]
 => 0
[4,1,1,1]
 => 3
[3,3,1]
 => 1
[3,2,2]
 => 2
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 4
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => 6
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
(load all 2 compositions to match this statistic)
Mapping
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 2
[4]
 => 0
[3,1]
 => 1
[2,2]
 => 2
[2,1,1]
 => 2
[1,1,1,1]
 => 3
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 2
[3,1,1]
 => 2
[2,2,1]
 => 3
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 4
[6]
 => 0
[5,1]
 => 1
[4,2]
 => 2
[4,1,1]
 => 2
[3,3]
 => 3
[3,2,1]
 => 3
[3,1,1,1]
 => 3
[2,2,2]
 => 4
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 5
[7]
 => 0
[6,1]
 => 1
[5,2]
 => 2
[5,1,1]
 => 2
[4,3]
 => 3
[4,2,1]
 => 3
[4,1,1,1]
 => 3
[3,3,1]
 => 4
[3,2,2]
 => 4
[3,2,1,1]
 => 4
[3,1,1,1,1]
 => 4
[2,2,2,1]
 => 5
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => 6
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
(load all 20 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => []
 => 0
[2]
 => []
 => 0
[1,1]
 => [1]
 => 1
[3]
 => []
 => 0
[2,1]
 => [1]
 => 1
[1,1,1]
 => [1,1]
 => 2
[4]
 => []
 => 0
[3,1]
 => [1]
 => 1
[2,2]
 => [2]
 => 2
[2,1,1]
 => [1,1]
 => 2
[1,1,1,1]
 => [1,1,1]
 => 3
[5]
 => []
 => 0
[4,1]
 => [1]
 => 1
[3,2]
 => [2]
 => 2
[3,1,1]
 => [1,1]
 => 2
[2,2,1]
 => [2,1]
 => 3
[2,1,1,1]
 => [1,1,1]
 => 3
[1,1,1,1,1]
 => [1,1,1,1]
 => 4
[6]
 => []
 => 0
[5,1]
 => [1]
 => 1
[4,2]
 => [2]
 => 2
[4,1,1]
 => [1,1]
 => 2
[3,3]
 => [3]
 => 3
[3,2,1]
 => [2,1]
 => 3
[3,1,1,1]
 => [1,1,1]
 => 3
[2,2,2]
 => [2,2]
 => 4
[2,2,1,1]
 => [2,1,1]
 => 4
[2,1,1,1,1]
 => [1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[7]
 => []
 => 0
[6,1]
 => [1]
 => 1
[5,2]
 => [2]
 => 2
[5,1,1]
 => [1,1]
 => 2
[4,3]
 => [3]
 => 3
[4,2,1]
 => [2,1]
 => 3
[4,1,1,1]
 => [1,1,1]
 => 3
[3,3,1]
 => [3,1]
 => 4
[3,2,2]
 => [2,2]
 => 4
[3,2,1,1]
 => [2,1,1]
 => 4
[3,1,1,1,1]
 => [1,1,1,1]
 => 4
[2,2,2,1]
 => [2,2,1]
 => 5
[2,2,1,1,1]
 => [2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000394
(load all 7 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => 0
[2]
 => [1,0,1,0]
 => 0
[1,1]
 => [1,1,0,0]
 => 1
[3]
 => [1,0,1,0,1,0]
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => 2
[4]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,2]
 => [1,1,1,0,0,0]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 3
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => 4
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 3
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => 2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 4
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => 4
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => 4
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 5
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [[1,2]]
 => 2 = 1 + 1
[1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => 3 = 2 + 1
[2,1]
 => [[1,2],[3]]
 => 2 = 1 + 1
[1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[3,1]
 => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[2,1,1]
 => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[4,1]
 => [[1,2,3,4],[5]]
 => 4 = 3 + 1
[3,2]
 => [[1,2,3],[4,5]]
 => 4 = 3 + 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => 3 = 2 + 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[6]
 => [[1,2,3,4,5,6]]
 => 6 = 5 + 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => 5 = 4 + 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => 5 = 4 + 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 4 = 3 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => 5 = 4 + 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 4 = 3 + 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => 3 = 2 + 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 4 = 3 + 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1 = 0 + 1
[7]
 => [[1,2,3,4,5,6,7]]
 => 7 = 6 + 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => 6 = 5 + 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => 6 = 5 + 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 5 = 4 + 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => 6 = 5 + 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => 5 = 4 + 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 4 = 3 + 1
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 5 = 4 + 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 5 = 4 + 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => 3 = 2 + 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 4 = 3 + 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1 = 0 + 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: St000738
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [[1,2]]
 => 1 = 0 + 1
[1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[3]
 => [[1,2,3]]
 => 1 = 0 + 1
[2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[3,2]
 => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
[6]
 => [[1,2,3,4,5,6]]
 => 1 = 0 + 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => 2 = 1 + 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => 3 = 2 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 3 = 2 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => 4 = 3 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 4 = 3 + 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 4 = 3 + 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 5 = 4 + 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 5 = 4 + 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 5 = 4 + 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[7]
 => [[1,2,3,4,5,6,7]]
 => 1 = 0 + 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => 2 = 1 + 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 2 + 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 3 = 2 + 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => 4 = 3 + 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 4 = 3 + 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 4 = 3 + 1
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 5 = 4 + 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 5 = 4 + 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 5 = 4 + 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 5 = 4 + 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 6 = 5 + 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 6 = 5 + 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 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: St000074
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [[1]]
 => 0
[2]
 => [[1,2]]
 => [[2,0],[1]]
 => 1
[1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 0
[3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2
[2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 0
[4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0
[5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
[4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0
[6]
 => [[1,2,3,4,5,6]]
 => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 5
[5,1]
 => [[1,2,3,4,5],[6]]
 => [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
[4,2]
 => [[1,2,3,4],[5,6]]
 => [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 4
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [[7,0,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 6
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [[6,1,0,0,0,0,0],[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 5
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [[5,2,0,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 5
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [[5,1,1,0,0,0,0],[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 5
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [[4,2,1,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [[4,1,1,1,0,0,0],[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [[3,3,1,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 4
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 4
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [[3,2,1,1,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [[3,1,1,1,1,0,0],[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 3
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [[2,2,1,1,1,0,0],[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [[2,1,1,1,1,1,0],[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[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],[1,1],[1]]
 => 0
Description
The number of special entries. An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Matching statistic: St000141
(load all 8 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => 0
[2]
 => [[1,2]]
 => [1,2] => 0
[1,1]
 => [[1],[2]]
 => [2,1] => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => 0
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 3
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 0
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 2
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 3
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 3
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 4
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 0
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 3
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => 3
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 4
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => 4
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => 4
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 5
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => 0
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => 2
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => 2
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => 3
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => 3
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => 3
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => 4
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => 4
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => 4
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => 4
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => 5
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => 5
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => 5
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => 6
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [[1]]
 => 0
[2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
[3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
[2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
[4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
[3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 2
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
[4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 3
[3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 3
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 2
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
[6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
[5,1]
 => [[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 4
[4,2]
 => [[1,2,3,4],[5,6]]
 => [[1,5],[2,6],[3],[4]]
 => 4
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [[1,5,6],[2],[3],[4]]
 => 3
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 4
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[1,4,6],[2,5],[3]]
 => 3
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 3
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 2
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 5
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [[1,6],[2,7],[3],[4],[5]]
 => 5
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [[1,6,7],[2],[3],[4],[5]]
 => 4
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [[1,5],[2,6],[3,7],[4]]
 => 5
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [[1,5,7],[2,6],[3],[4]]
 => 4
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [[1,5,6,7],[2],[3],[4]]
 => 3
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [[1,4,7],[2,5],[3,6]]
 => 4
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [[1,4,6],[2,5,7],[3]]
 => 4
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [[1,4,6,7],[2,5],[3]]
 => 3
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [[1,4,5,6,7],[2],[3]]
 => 2
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => 3
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [[1,3,5,6,7],[2,4]]
 => 2
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 0
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000211
(load all 25 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => {{1}}
 => 0
[2]
 => [[1,2]]
 => {{1,2}}
 => 1
[1,1]
 => [[1],[2]]
 => {{1},{2}}
 => 0
[3]
 => [[1,2,3]]
 => {{1,2,3}}
 => 2
[2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0
[4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3
[3,1]
 => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2
[2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0
[5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4
[4,1]
 => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => 3
[3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 3
[3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 2
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 0
[6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 5
[5,1]
 => [[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => 4
[4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => 4
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => 3
[3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => 4
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => 3
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => 3
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => {{1,6},{2},{3},{4},{5}}
 => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => {{1,2,3,4,5,6,7}}
 => 6
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => {{1,3,4,5,6,7},{2}}
 => 5
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => {{1,2,5,6,7},{3,4}}
 => 5
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => {{1,4,5,6,7},{2},{3}}
 => 4
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => {{1,2,3,7},{4,5,6}}
 => 5
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => {{1,3,6,7},{2,5},{4}}
 => 4
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => 3
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => {{1,3,4},{2,6,7},{5}}
 => 4
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => {{1,2,7},{3,4},{5,6}}
 => 4
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => {{1,4,7},{2,6},{3},{5}}
 => 3
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => {{1,6,7},{2},{3},{4},{5}}
 => 2
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => {{1,3},{2,5},{4,7},{6}}
 => 3
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => {{1,5},{2,7},{3},{4},{6}}
 => 2
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => {{1,7},{2},{3},{4},{5},{6}}
 => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => 0
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
The following 310 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000376The bounce deficit of a Dyck path. St000441The number of successions 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. St001034The area of the parallelogram polyomino associated with the Dyck path. St000054The first entry of the permutation. St000734The last entry in the first row of a standard tableau. St000839The largest opener of a set partition. St000018The number of inversions of a permutation. St000024The number of double up and double down steps of a Dyck path. St000081The number of edges of a graph. St000171The degree of the graph. St000246The number of non-inversions of a permutation. St000272The treewidth of a graph. St000290The major index of a binary word. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001485The modular major index of a binary word. St000010The length of the partition. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000505The biggest entry in the block containing the 1. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000971The smallest closer of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of 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. St000369The dinv deficit of a Dyck path. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000288The number of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000503The maximal difference between two elements in a common block. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000925The number of topologically connected components of a set partition. St001062The maximal size of a block of a set partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000209Maximum difference of elements in cycles. St001298The number of repeated entries in the Lehmer code of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000539The number of odd inversions of a permutation. St000795The mad of a permutation. St000833The comajor index of a permutation. St000956The maximal displacement of a permutation. St000654The first descent of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000019The cardinality of the support of a permutation. St001277The degeneracy of a graph. St000740The last entry of a permutation. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000029The depth of a permutation. St000224The sorting index of a permutation. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000004The major index of a permutation. St000030The sum of the descent differences of a permutations. St000189The number of elements in the poset. St000214The number of adjacencies of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000395The sum of the heights of the peaks of a Dyck path. St000794The mak of a permutation. St000798The makl of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001077The prefix exchange distance of a 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. 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. St001397Number of pairs of incomparable elements in a finite poset. St001428The number of B-inversions of a signed permutation. St001759The Rajchgot index of a permutation. St001869The maximum cut size of a graph. St000240The number of indices that are not small excedances. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001480The number of simple summands of the module J^2/J^3. St000454The largest eigenvalue of a graph if it is integral. St000670The reversal length of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000653The last descent of a permutation. St000719The number of alignments in a perfect matching. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001644The dimension of a graph. St000308The height of the tree associated to a permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001760The number of prefix or suffix reversals needed to sort a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001090The number of pop-stack-sorts needed to sort a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000021The number of descents of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001497The position of the largest weak excedence of a permutation. St000005The bounce statistic of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001082The number of boxed occurrences of 123 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St000060The greater neighbor of the maximum. 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. St001555The order of a signed permutation. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000327The number of cover relations in a poset. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001330The hat guessing number of a graph. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000739The first entry in the last row of a semistandard tableau. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. 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. St001118The acyclic chromatic index of a graph. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001821The sorting index of a signed permutation. St000120The number of left tunnels of a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001596The number of two-by-two squares inside a skew partition. 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. St000422The energy of a graph, if it is integral. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St000006The dinv of a Dyck path. St000766The number of inversions of an integer composition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000691The number of changes of a binary word. St001267The length of the Lyndon factorization of the binary word. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001935The number of ascents in a parking function. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. 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. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St001096The size of the overlap set of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000521The number of distinct subtrees of an ordered tree. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000091The descent variation of a composition. St000173The segment statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000360The number of occurrences of the pattern 32-1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000491The number of inversions of a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. 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. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000650The number of 3-rises of a permutation. St001060The distinguishing index of a graph. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001843The Z-index of a set partition. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001461The number of topologically connected components of the chord diagram of a permutation. St001645The pebbling number of a connected graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000456The monochromatic index of a connected graph. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000352The Elizalde-Pak rank of a permutation. St000356The number of occurrences of the pattern 13-2. St000834The number of right outer peaks of a permutation. St000007The number of saliances of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St000365The number of double ascents of a permutation. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000023The number of inner peaks of a permutation. St000090The variation of a composition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (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. St000624The normalized sum of the minimal distances to a greater element. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000779The tier of a permutation. St001469The holeyness 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001557The number of inversions of the second entry of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau.