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Your data matches 889 different statistics following compositions of up to 3 maps.
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Matching statistic: St000657
(load all 87 compositions to match this statistic)
(load all 87 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [1,1,1] => 1 = 3 - 2
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [1,1,1] => 1 = 3 - 2
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [1,1,1] => 1 = 3 - 2
Description
The smallest part of an integer composition.
Matching statistic: St000510
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000713
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
Description
The dimension of the irreducible representation of Sp(4) labelled by an integer partition.
Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$.
For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.
Matching statistic: St000714
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000929
(load all 148 compositions to match this statistic)
(load all 148 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001283
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [3,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [4,2,1]
=> 0 = 3 - 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [4,2,1]
=> 0 = 3 - 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [3,2,2]
=> 0 = 3 - 3
Description
The number of finite groups that are realised by the given partition over the complex numbers.
A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St000264
(load all 56 compositions to match this statistic)
(load all 56 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001261
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000326
(load all 112 compositions to match this statistic)
(load all 112 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,4,6],[2,5]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6],[2],[5]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4],[2,5,6]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,4],[3,5,6]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,4,6],[2,5],[3]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,6],[2,4],[5]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,6],[3,4],[5]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4],[2,6],[5]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,4],[3,6],[5]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3],[2,4],[5,6]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2],[3,4],[5,6]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6,7],[2,5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6,7],[2],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,7],[2,5,6]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6],[2,5,7]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,3,5],[4,6,7]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,6,7],[2,4],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,7],[2,6],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,7],[2,5],[6]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,3,7],[4,5],[6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6],[2,7],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,5],[2,7],[6]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6],[2,5],[7]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,6,7],[2],[4],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,7],[2],[5],[6]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4,6],[2],[5],[7]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,6],[2,4,7],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,5],[3,6,7],[4]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4],[2,6,7],[5]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4],[2,5,7],[6]]
=> [2,4,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,3],[4,5,7],[6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4],[2,5,6],[7]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,2,7],[3,5],[4,6]]
=> [4,2,1] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,7],[2,4],[5,6]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,6],[2,4],[5,7]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
[[1,3,4],[2,6],[5,7]]
=> [2,3,2] => [1,1,1] => 111 => 1 = 3 - 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
The following 879 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000535The rank-width of a graph. St000667The greatest common divisor of the parts of the partition. St000775The multiplicity of the largest eigenvalue in a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000805The number of peaks of the associated bargraph. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001272The number of graphs with the same degree sequence. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001568The smallest positive integer that does not appear twice in the partition. St001597The Frobenius rank of a skew partition. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000478Another weight of a partition according to Alladi. St000629The defect of a binary word. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001577The minimal number of edges to add or remove to make a graph a cograph. St001593This is the number of standard Young tableaux of the given shifted shape. St001596The number of two-by-two squares inside a skew partition. St001793The difference between the clique number and the chromatic number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000759The smallest missing part in an integer partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000378The diagonal inversion number of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000995The largest even part of an integer partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000480The number of lower covers of a partition in dominance order. St000913The number of ways to refine the partition into singletons. St001092The number of distinct even parts of a partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000477The weight of a partition according to Alladi. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001490The number of connected components of a skew partition. St000920The logarithmic height of a Dyck path. St000296The length of the symmetric border of a binary word. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000052The number of valleys of a Dyck path not on the x-axis. 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000481The number of upper covers of a partition in dominance order. St001484The number of singletons of an integer partition. St000260The radius of a connected graph. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001696The natural major index of a standard Young tableau. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000183The side length of the Durfee square of an integer partition. St001814The number of partitions interlacing the given partition. St000257The number of distinct parts of a partition that occur at least twice. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000297The number of leading ones in a binary word. St000627The exponent of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001722The number of minimal chains with small intervals between a binary word and the top element. St000047The number of standard immaculate tableaux of a given shape. St000993The multiplicity of the largest part of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000733The row containing the largest entry of a standard tableau. St000011The number of touch points (or returns) of a Dyck path. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000143The largest repeated part of a partition. St000783The side length of the largest staircase partition fitting into a partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001571The Cartan determinant of the integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000382The first part of an integer composition. St000842The breadth of a permutation. St000115The single entry in the last row. St000651The maximal size of a rise in a permutation. St000655The length of the minimal rise of a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in 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. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001394The genus of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000678The number of up steps after the last double rise of a Dyck path. St000701The protection number of a binary tree. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St001256Number of simple reflexive modules that are 2-stable reflexive. St000392The length of the longest run of ones in a binary word. St000381The largest part of an integer composition. St000764The number of strong records in an integer composition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000659The number of rises of length at least 2 of a Dyck path. St001732The number of peaks visible from the left. St000253The crossing number of a set partition. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000862The number of parts of the shifted shape of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000877The depth of the binary word interpreted as a path. St000057The Shynar inversion number of a standard tableau. St000397The Strahler number of a rooted tree. St000068The number of minimal elements in a poset. St000876The number of factors in the Catalan decomposition of a binary word. St000042The number of crossings of a perfect matching. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000974The length of the trunk of an ordered tree. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000897The number of different multiplicities of parts of an integer partition. St001280The number of parts of an integer partition that are at least two. St001175The size of a partition minus the hook length of the base cell. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000284The Plancherel distribution on integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000517The Kreweras number of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000935The number of ordered refinements of an integer partition. St001129The product of the squares of the parts of a partition. St001389The number of partitions of the same length below the given integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000142The number of even parts of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001214The aft of an integer partition. St001248Sum of the even parts of a partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St000390The number of runs of ones in a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000383The last part of an integer composition. St000439The position of the first down step of a Dyck path. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000234The number of global ascents of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000903The number of different parts of an integer composition. St001730The number of times the path corresponding to a binary word crosses the base line. St000905The number of different multiplicities of parts of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000731The number of double exceedences of a permutation. St000288The number of ones in a binary word. St000292The number of ascents of a binary word. St000389The number of runs of ones of odd length in a binary word. St000628The balance of a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000069The number of maximal elements of a poset. St001741The largest integer such that all patterns of this size are contained in the permutation. 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. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000007The number of saliances of the permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000451The length of the longest pattern of the form k 1 2. St000374The number of exclusive right-to-left minima of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000078The number of alternating sign matrices whose left key is the permutation. St000761The number of ascents in an integer composition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000732The number of double deficiencies of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000542The number of left-to-right-minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000056The decomposition (or block) number 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. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001381The fertility of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001663The number of occurrences of the Hertzsprung pattern 132 in 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. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000660The number of rises of length at least 3 of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001665The number of pure excedances of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000989The number of final rises of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000990The first ascent of a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000444The length of the maximal rise of a Dyck path. St000904The maximal number of repetitions of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000847The number of standard Young tableaux whose descent set is the binary word. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000347The inversion sum of a binary word. St000644The number of graphs with given frequency partition. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St000013The height of a Dyck path. St000306The bounce count of a Dyck path. St001111The weak 2-dynamic chromatic number of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001561The value of the elementary symmetric function evaluated at 1. St000679The pruning number of an ordered tree. St000396The register function (or Horton-Strahler number) of a binary tree. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000160The multiplicity of the smallest part of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St001933The largest multiplicity of a part in an integer partition. St000150The floored half-sum of the multiplicities of a partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000983The length of the longest alternating subword. St001313The number of Dyck paths above the lattice path given by a binary word. St001838The number of nonempty primitive factors of a binary word. St001885The number of binary words with the same proper border set. St000289The decimal representation of a binary word. St000348The non-inversion sum of a binary word. St000391The sum of the positions of the ones in a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000386The number of factors DDU in a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000765The number of weak records in an integer composition. St000899The maximal number of repetitions of an integer composition. St001413Half the length of the longest even length palindromic prefix of a binary word. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000816The number of standard composition tableaux of the composition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000002The number of occurrences of the pattern 123 in a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St000735The last entry on the main diagonal of a standard tableau. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000053The number of valleys of the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000658The number of rises of length 2 of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000017The number of inversions of a standard tableau. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St001330The hat guessing number of a graph. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000666The number of right tethers of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. 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. 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. St000312The number of leaves in a graph. St000617The number of global maxima of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000352The Elizalde-Pak rank of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000091The descent variation of a composition. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000054The first entry of the permutation. St000359The number of occurrences of the pattern 23-1. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000932The number of occurrences of the pattern UDU in a Dyck path. St000729The minimal arc length of a set partition. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. 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. St000883The number of longest increasing subsequences of a permutation. St000015The number of peaks of a Dyck path. St000331The number of upper interactions of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001530The depth of a Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St000271The chromatic index of a graph. St000058The order of a permutation. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000065The number of entries equal to -1 in an alternating sign matrix. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000919The number of maximal left branches of a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000298The order dimension or Dushnik-Miller dimension of a poset. St001090The number of pop-stack-sorts needed to sort a permutation. St000647The number of big descents of a permutation. St001050The number of terminal closers of a set partition. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000823The number of unsplittable factors of the set partition. St000971The smallest closer of a set partition. St001075The minimal size of a block of a set partition. St000028The number of stack-sorts needed to sort a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000402Half the size of the symmetry class of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St000097The order of the largest clique of the graph. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000475The number of parts equal to 1 in a partition. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000463The number of admissible inversions of a permutation. St001737The number of descents of type 2 in a permutation. St000355The number of occurrences of the pattern 21-3. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000700The protection number of an ordered tree. St001468The smallest fixpoint of a permutation. St000210Minimum over maximum difference of elements in cycles. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000648The number of 2-excedences of a permutation. St000254The nesting number of a set partition. St000730The maximal arc length of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent 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. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. 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. St000748The major index of the permutation obtained by flattening the set partition. St000141The maximum drop size of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001270The bandwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001471The magnitude of a Dyck path. St001494The Alon-Tarsi number of a graph. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000258The burning number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000637The length of the longest cycle in a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001110The 3-dynamic chromatic number of a graph. St001120The length of a longest path in a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001458The rank of the adjacency matrix of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001689The number of celebrities in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001716The 1-improper chromatic number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001792The arboricity of a graph. St000363The number of minimal vertex covers of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000552The number of cut vertices of a graph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001368The number of vertices of maximal degree in a graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001743The discrepancy of a graph. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000268The number of strongly connected orientations of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001797The number of overfull subgraphs of a graph. St001563The value of the power-sum symmetric function evaluated at 1. St000071The number of maximal chains in a poset. St000098The chromatic number of a graph. St000171The degree of the graph. St000527The width of the poset. St000568The hook number of a binary tree. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001486The number of corners of the ribbon associated with an integer composition. St001512The minimum rank of a graph. St001667The maximal size of a pair of weak twins for a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000024The number of double up and double down steps of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000259The diameter of a connected graph. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000537The cutwidth of a graph. St000632The jump number of the poset. St000767The number of runs in an integer composition. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St000846The maximal number of elements covering an element of a poset. St000918The 2-limited packing number of a graph. St000991The number of right-to-left minima of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001062The maximal size of a block of a set partition. St001109The number of proper colourings of a graph with as few colours as possible. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a graph. 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. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St001962The proper pathwidth of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000273The domination number of a graph. St000456The monochromatic index of a connected graph. St000544The cop number of a graph. St000654The first descent of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000722The number of different neighbourhoods in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000845The maximal number of elements covered by an element in a poset. St000907The number of maximal antichains of minimal length in a poset. St000909The number of maximal chains of maximal size in a poset. St000916The packing number of a graph. St000948The chromatic discriminant of a graph. St001271The competition number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001546The number of monomials in the Tutte polynomial of a graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001592The maximal number of simple paths between any two different vertices of a graph. St001638The book thickness of a graph. St001673The degree of asymmetry of an integer composition. St001694The number of maximal dissociation sets in a graph. St001729The number of visible descents of a permutation. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001829The common independence number of a graph. St001871The number of triconnected components of a graph. St000090The variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000322The skewness of a graph. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000403The Szeged index minus the Wiener index of a graph. St000516The number of stretching pairs of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000664The number of right ropes of a permutation. St000699The toughness times the least common multiple of 1,. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000962The 3-shifted major index of a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001119The length of a shortest maximal path in a graph. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001736The total number of cycles in a graph. St001781The interlacing number of a set partition. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001826The maximal number of leaves on a vertex of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. 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 $eA$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001618The cardinality of the Frattini sublattice of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001623The number of doubly irreducible elements of a lattice. St001301The first Betti number of the order complex associated with the poset. St001307The number of induced stars on four vertices in a graph. St000886The number of permutations with the same antidiagonal sums. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St000479The Ramsey number of a graph. St001060The distinguishing index of a graph. St001734The lettericity of a graph. St001725The harmonious chromatic number of a graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000511The number of invariant subsets when acting with a permutation of given cycle type. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000649The number of 3-excedences of a permutation. St000710The number of big deficiencies of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000455The second largest eigenvalue of a graph if it is integral. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000717The number of ordinal summands of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000961The shifted major index of a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001717The largest size of an interval in a poset. St000640The rank of the largest boolean interval in a poset. St001718The number of non-empty open intervals in a poset. St001902The number of potential covers of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001472The permanent of the Coxeter matrix of the poset. St001345The Hamming dimension of a graph. St001469The holeyness of a permutation. St001479The number of bridges of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000461The rix statistic of a permutation. St000539The number of odd inversions of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001735The number of permutations with the same set of runs. St001801Half the number of preimage-image pairs of different parity in a permutation. St001928The number of non-overlapping descents in a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St001727The number of invisible inversions of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001260The permanent of an alternating sign matrix. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001834The number of non-isomorphic minors of a graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001081The number of minimal length factorizations of a permutation into star transpositions. St000401The size of the symmetry class of a permutation. St000467The hyper-Wiener index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000464The Schultz index of a connected graph. St001875The number of simple modules with projective dimension at most 1. 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. St000673The number of non-fixed points of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000873The aix statistic of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001114The number of odd descents of a permutation. St000956The maximal displacement of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001566The length of the longest arithmetic progression in a permutation.
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