Your data matches 463 different statistics following compositions of up to 3 maps.
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Matching statistic: St000225
(load all 3 compositions to match this statistic)
Mapping
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 2
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 3
[3,2]
 => 1
[3,1,1]
 => 2
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 4
[4,2]
 => 2
[4,1,1]
 => 3
[3,3]
 => 0
[3,2,1]
 => 2
[3,1,1,1]
 => 2
[2,2,2]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 5
[5,2]
 => 3
[5,1,1]
 => 4
[4,3]
 => 1
[4,2,1]
 => 3
[4,1,1,1]
 => 3
[3,3,1]
 => 2
[3,2,2]
 => 1
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 2
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001804
(load all 2 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: 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]]
 => 1 = 0 + 1
[3]
 => [[1,2,3]]
 => 1 = 0 + 1
[2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[2,2]
 => [[1,2],[3,4]]
 => 1 = 0 + 1
[2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 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]]
 => 2 = 1 + 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 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]]
 => 2 = 1 + 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 3 = 2 + 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => 1 = 0 + 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 3 = 2 + 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 4 = 3 + 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 1 = 0 + 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3 = 2 + 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]]
 => 1 = 0 + 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]]
 => 2 = 1 + 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 3 = 2 + 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => 2 = 1 + 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 3 = 2 + 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]]
 => 2 = 1 + 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 3 = 2 + 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 4 = 3 + 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]]
 => 2 = 1 + 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 4 = 3 + 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]]
 => 1 = 0 + 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St000089
(load all 2 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000089: Integer compositions ⟶ 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],[2]]
 => [1,1] => 0
[3]
 => [[1,2,3]]
 => [3] => 0
[2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 0
[4]
 => [[1,2,3,4]]
 => [4] => 0
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => 2
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 0
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 0
[5]
 => [[1,2,3,4,5]]
 => [5] => 0
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 3
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 0
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 0
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 4
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 2
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 3
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 0
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 2
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 0
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 0
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => 5
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 3
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => 4
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => 3
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => 3
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => 2
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => 2
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => 2
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 0
Description
The absolute variation of a composition.
Matching statistic: St000090
(load all 3 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000090: Integer compositions ⟶ 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],[2]]
 => [1,1] => 0
[3]
 => [[1,2,3]]
 => [3] => 0
[2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 0
[4]
 => [[1,2,3,4]]
 => [4] => 0
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => 2
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 0
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 0
[5]
 => [[1,2,3,4,5]]
 => [5] => 0
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 3
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 0
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 0
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 4
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 3
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 0
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 2
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 0
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 0
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [1,6] => 5
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => 3
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [1,1,5] => 4
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => 3
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [1,1,1,4] => 3
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => 2
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => 2
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [1,1,1,1,3] => 2
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [1,1,1,2,2] => 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,2] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 0
Description
The variation of a composition.
Matching statistic: St000091
(load all 5 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000091: Integer compositions ⟶ 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],[2]]
 => [1,1] => 0
[3]
 => [[1,2,3]]
 => [3] => 0
[2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 0
[4]
 => [[1,2,3,4]]
 => [4] => 0
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => 2
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 0
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 0
[5]
 => [[1,2,3,4,5]]
 => [5] => 0
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 3
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 2
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 0
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 0
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 4
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 3
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 0
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 2
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 0
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 0
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [1,6] => 5
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => 3
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [1,1,5] => 4
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => 3
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [1,1,1,4] => 3
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [1,3,3] => 2
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [2,2,3] => 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [1,1,2,3] => 2
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [1,1,1,1,3] => 2
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [1,2,2,2] => 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [1,1,1,2,2] => 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,2] => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 0
Description
The descent variation of a composition. Defined in [1].
Matching statistic: St000010
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1],[]]
 => [[1],[]]
 => []
 => 0
[2]
 => [[2],[]]
 => [[2],[]]
 => []
 => 0
[1,1]
 => [[1,1],[]]
 => [[1,1],[]]
 => []
 => 0
[3]
 => [[3],[]]
 => [[3],[]]
 => []
 => 0
[2,1]
 => [[2,1],[]]
 => [[2,2],[1]]
 => [1]
 => 1
[1,1,1]
 => [[1,1,1],[]]
 => [[1,1,1],[]]
 => []
 => 0
[4]
 => [[4],[]]
 => [[4],[]]
 => []
 => 0
[3,1]
 => [[3,1],[]]
 => [[3,3],[2]]
 => [2]
 => 1
[2,2]
 => [[2,2],[]]
 => [[2,2],[]]
 => []
 => 0
[2,1,1]
 => [[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 2
[1,1,1,1]
 => [[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => []
 => 0
[5]
 => [[5],[]]
 => [[5],[]]
 => []
 => 0
[4,1]
 => [[4,1],[]]
 => [[4,4],[3]]
 => [3]
 => 1
[3,2]
 => [[3,2],[]]
 => [[3,3],[1]]
 => [1]
 => 1
[3,1,1]
 => [[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[2,2,1]
 => [[2,2,1],[]]
 => [[2,2,2],[1]]
 => [1]
 => 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => [[1,1,1,1,1],[]]
 => []
 => 0
[6]
 => [[6],[]]
 => [[6],[]]
 => []
 => 0
[5,1]
 => [[5,1],[]]
 => [[5,5],[4]]
 => [4]
 => 1
[4,2]
 => [[4,2],[]]
 => [[4,4],[2]]
 => [2]
 => 1
[4,1,1]
 => [[4,1,1],[]]
 => [[4,4,4],[3,3]]
 => [3,3]
 => 2
[3,3]
 => [[3,3],[]]
 => [[3,3],[]]
 => []
 => 0
[3,2,1]
 => [[3,2,1],[]]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
[2,2,2]
 => [[2,2,2],[]]
 => [[2,2,2],[]]
 => []
 => 0
[2,2,1,1]
 => [[2,2,1,1],[]]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 2
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1],[]]
 => []
 => 0
[7]
 => [[7],[]]
 => [[7],[]]
 => []
 => 0
[6,1]
 => [[6,1],[]]
 => [[6,6],[5]]
 => [5]
 => 1
[5,2]
 => [[5,2],[]]
 => [[5,5],[3]]
 => [3]
 => 1
[5,1,1]
 => [[5,1,1],[]]
 => [[5,5,5],[4,4]]
 => [4,4]
 => 2
[4,3]
 => [[4,3],[]]
 => [[4,4],[1]]
 => [1]
 => 1
[4,2,1]
 => [[4,2,1],[]]
 => [[4,4,4],[3,2]]
 => [3,2]
 => 2
[4,1,1,1]
 => [[4,1,1,1],[]]
 => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => 3
[3,3,1]
 => [[3,3,1],[]]
 => [[3,3,3],[2]]
 => [2]
 => 1
[3,2,2]
 => [[3,2,2],[]]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 2
[3,2,1,1]
 => [[3,2,1,1],[]]
 => [[3,3,3,3],[2,2,1]]
 => [2,2,1]
 => 3
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => 4
[2,2,2,1]
 => [[2,2,2,1],[]]
 => [[2,2,2,2],[1]]
 => [1]
 => 1
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => [[1,1,1,1,1,1,1],[]]
 => []
 => 0
Description
The length of the partition.
Matching statistic: St000052
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [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]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [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]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 5
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 3
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => 4
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 3
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000147
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1],[]]
 => [[1],[]]
 => []
 => 0
[2]
 => [[2],[]]
 => [[2],[]]
 => []
 => 0
[1,1]
 => [[1,1],[]]
 => [[1,1],[]]
 => []
 => 0
[3]
 => [[3],[]]
 => [[3],[]]
 => []
 => 0
[2,1]
 => [[2,1],[]]
 => [[2,2],[1]]
 => [1]
 => 1
[1,1,1]
 => [[1,1,1],[]]
 => [[1,1,1],[]]
 => []
 => 0
[4]
 => [[4],[]]
 => [[4],[]]
 => []
 => 0
[3,1]
 => [[3,1],[]]
 => [[3,3],[2]]
 => [2]
 => 2
[2,2]
 => [[2,2],[]]
 => [[2,2],[]]
 => []
 => 0
[2,1,1]
 => [[2,1,1],[]]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[1,1,1,1]
 => [[1,1,1,1],[]]
 => [[1,1,1,1],[]]
 => []
 => 0
[5]
 => [[5],[]]
 => [[5],[]]
 => []
 => 0
[4,1]
 => [[4,1],[]]
 => [[4,4],[3]]
 => [3]
 => 3
[3,2]
 => [[3,2],[]]
 => [[3,3],[1]]
 => [1]
 => 1
[3,1,1]
 => [[3,1,1],[]]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[2,2,1]
 => [[2,2,1],[]]
 => [[2,2,2],[1]]
 => [1]
 => 1
[2,1,1,1]
 => [[2,1,1,1],[]]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => [[1,1,1,1,1],[]]
 => []
 => 0
[6]
 => [[6],[]]
 => [[6],[]]
 => []
 => 0
[5,1]
 => [[5,1],[]]
 => [[5,5],[4]]
 => [4]
 => 4
[4,2]
 => [[4,2],[]]
 => [[4,4],[2]]
 => [2]
 => 2
[4,1,1]
 => [[4,1,1],[]]
 => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[3,3]
 => [[3,3],[]]
 => [[3,3],[]]
 => []
 => 0
[3,2,1]
 => [[3,2,1],[]]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
[2,2,2]
 => [[2,2,2],[]]
 => [[2,2,2],[]]
 => []
 => 0
[2,2,1,1]
 => [[2,2,1,1],[]]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => [[2,2,2,2,2],[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
[7]
 => [[7],[]]
 => [[7],[]]
 => []
 => 0
[6,1]
 => [[6,1],[]]
 => [[6,6],[5]]
 => [5]
 => 5
[5,2]
 => [[5,2],[]]
 => [[5,5],[3]]
 => [3]
 => 3
[5,1,1]
 => [[5,1,1],[]]
 => [[5,5,5],[4,4]]
 => [4,4]
 => 4
[4,3]
 => [[4,3],[]]
 => [[4,4],[1]]
 => [1]
 => 1
[4,2,1]
 => [[4,2,1],[]]
 => [[4,4,4],[3,2]]
 => [3,2]
 => 3
[4,1,1,1]
 => [[4,1,1,1],[]]
 => [[4,4,4,4],[3,3,3]]
 => [3,3,3]
 => 3
[3,3,1]
 => [[3,3,1],[]]
 => [[3,3,3],[2]]
 => [2]
 => 2
[3,2,2]
 => [[3,2,2],[]]
 => [[3,3,3],[1,1]]
 => [1,1]
 => 1
[3,2,1,1]
 => [[3,2,1,1],[]]
 => [[3,3,3,3],[2,2,1]]
 => [2,2,1]
 => 2
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => [[3,3,3,3,3],[2,2,2,2]]
 => [2,2,2,2]
 => 2
[2,2,2,1]
 => [[2,2,2,1],[]]
 => [[2,2,2,2],[1]]
 => [1]
 => 1
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => [[2,2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => [[2,2,2,2,2,2],[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],[]]
 => [[1,1,1,1,1,1,1],[]]
 => []
 => 0
Description
The largest part of an integer partition.
Matching statistic: St000204
(load all 2 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [.,.]
 => 0
[2]
 => [1,0,1,0]
 => [1,1,0,0]
 => [[.,.],.]
 => 0
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [.,[.,.]]
 => 0
[3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [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]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => 3
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => 0
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => 5
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => 3
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[[[[[.,.],.],.],.],.]]
 => 4
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[[[.,.],.],.]]
 => 3
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [[[.,[.,.]],.],[[.,.],.]]
 => 2
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [.,[[.,[.,.]],[.,.]]]
 => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => 0
Description
The number of internal nodes of a binary tree. That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000356
(load all 5 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1] => [1] => 0
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0
[1,1]
 => [1,1,0,0]
 => [2,1] => [2,1] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 0
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 1
[2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,3,2,1] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,4,3,1] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 3
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,4,5,3] => 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,3,4,1] => 0
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,3,5,4,2] => 2
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,6,5,4,3] => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [3,2,4,1] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [2,5,4,3,1] => 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,6,5,4,3,2] => 4
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6,5,4,3,2,1] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,3,4,5,2] => 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,4,6,5,3] => 2
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => 3
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,5,4,1] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,4,3,5,2] => 2
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [1,3,6,5,4,2] => 3
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,6,5,4,3] => 4
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [3,2,5,4,1] => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => [2,6,5,4,3,1] => 3
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => 5
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => 0
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
The following 453 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000446The disorder of a permutation. St000496The rcs statistic of a set partition. St000766The number of inversions of an integer composition. St000996The number of exclusive left-to-right maxima 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. St000288The number of ones in a binary word. St000691The number of changes of a binary word. St000028The number of stack-sorts needed to sort a permutation. St001323The independence gap of a graph. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000358The number of occurrences of the pattern 31-2. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in 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. 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. 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. St000155The number of exceedances (also excedences) of a permutation. St000317The cycle descent number of a permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000710The number of big deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000451The length of the longest pattern of the form k 1 2. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000503The maximal difference between two elements in a common block. St000653The last descent of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000956The maximal displacement of a 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). St000355The number of occurrences of the pattern 21-3. St001520The number of strict 3-descents. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001742The difference of the maximal and the minimal degree in a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000145The Dyson rank of a partition. St000365The number of double ascents of a permutation. St000462The major index minus the number of excedences of 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 $K[x]/(x^n)$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001513The number of nested exceedences of a permutation. St001689The number of celebrities in a graph. St000886The number of permutations with the same antidiagonal sums. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001674The number of vertices of the largest induced star graph in the graph. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000148The number of odd parts of a partition. St000228The size of a partition. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000335The difference of lower and upper interactions. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001360The number of covering relations in Young's lattice below a partition. St001372The length of a longest cyclic run of ones of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001523The degree of symmetry of a 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. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001959The product of the heights of the peaks of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001867The number of alignments of type EN of a signed permutation. St001964The interval resolution global dimension of a poset. 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. St000177The number of free tiles in the pattern. St000178Number of free entries. St000454The largest eigenvalue of a graph if it is integral. St001438The number of missing boxes of a skew partition. St001846The number of elements which do not have a complement in the lattice. St001487The number of inner corners of a skew partition. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000769The major index of a composition regarded as a word. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000478Another weight of a partition according to Alladi. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St001820The size of the image of the pop stack sorting operator. St000422The energy of a graph, if it is integral. St001556The number of inversions of the third entry of a permutation. St000516The number of stretching pairs of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001822The number of alignments of a signed permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000299The number of nonisomorphic vertex-induced subtrees. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000488The number of cycles of a permutation of length at most 2. St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001856The number of edges in the reduced word graph of a permutation. St001868The number of alignments of type NE of a signed permutation. St001948The number of augmented double ascents of a permutation. St000456The monochromatic index of a connected graph. St000902 The minimal number of repetitions of an integer composition. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001530The depth of a Dyck path. St000527The width of the poset. St000455The second largest eigenvalue of a graph if it is integral. St000567The sum of the products of all pairs of parts. 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. St000707The product of the factorials of the parts. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St001866The nesting alignments of a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000068The number of minimal elements in a poset. St000534The number of 2-rises of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St000842The breadth of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000679The pruning number of an ordered tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000217The number of occurrences of the pattern 312 in a permutation. St000233The number of nestings of a set partition. St000284The Plancherel distribution on integer partitions. St000338The number of pixed points of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000650The number of 3-rises of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000709The number of occurrences of 14-2-3 or 14-3-2. St000779The tier of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001896The number of right descents of a signed permutations. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000546The number of global descents of a permutation. 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. St000731The number of double exceedences of a permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St001060The distinguishing index of a graph. 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. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001490The number of connected components of a skew partition. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000862The number of parts of the shifted shape of a permutation. St001488The number of corners of a skew partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001875The number of simple modules with projective dimension at most 1. St000039The number of crossings of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000666The number of right tethers of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000872The number of very big descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001301The first Betti number of the order complex associated with the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000210Minimum over maximum difference of elements in cycles. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys 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. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000646The number of big ascents of a permutation. St000654The first descent of a permutation. St000663The number of right floats of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001388The number of non-attacking neighbors of a permutation. St001399The distinguishing number of a poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001469The holeyness of a permutation. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001712The number of natural descents of a standard Young tableau. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000354The number of recoils of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000619The number of cyclic descents of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001569The maximal modular displacement of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St001424The number of distinct squares in a binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.