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Your data matches 259 different statistics following compositions of up to 3 maps.
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Matching statistic: St001520
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(load all 3 compositions to match this statistic)
St001520: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 1
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The number of strict 3-descents.
A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Matching statistic: St000538
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St000538: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St000538: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,-1] => [1,2] => 0
[2,1] => [2,1] => [1,-2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,-1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,-1] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1,2,3] => 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,1,-2] => [2,1,3] => 0
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1,3,2,4] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [2,1,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [3,1,2,4] => 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [2,3,1,4] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [3,2,1,4] => 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,2,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,3,2,4] => 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,3,4] => 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,4] => 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,2,4] => 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,1,3,4] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,3,4] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,4] => 0
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [3,1,2,4] => 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,4] => 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,2,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [2,3,1,4] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [2,3,1,4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,1,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1,2,4,3,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,4,2,3,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [1,3,4,2,5] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [1,4,3,2,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [2,1,3,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [3,1,2,4,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [4,1,2,3,5] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [3,1,4,2,5] => 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [4,1,3,2,5] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [2,3,1,4,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [2,4,1,3,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [3,2,1,4,5] => 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [4,2,1,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [3,4,1,2,5] => 2
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [4,3,1,2,5] => 2
Description
The number of even inversions of a permutation.
An inversion $i < j$ of a permutation is even if $i \equiv j~(\operatorname{mod} 2)$. See [[St000539]] for odd inversions.
Matching statistic: St000836
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(load all 2 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,-1] => [1,2] => 0
[2,1] => [2,1] => [1,-2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,-1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,-1] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1,2,3] => 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,1,-2] => [2,1,3] => 0
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1,3,2,4] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [2,1,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [3,1,2,4] => 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [2,3,1,4] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [3,2,1,4] => 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,2,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,3,2,4] => 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,3,4] => 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,4] => 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,2,4] => 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,1,3,4] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,3,4] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,4] => 0
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [3,1,2,4] => 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,4] => 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,2,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [2,3,1,4] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [2,3,1,4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,1,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1,2,4,3,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,4,2,3,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [1,3,4,2,5] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [1,4,3,2,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [2,1,3,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [3,1,2,4,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [4,1,2,3,5] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [3,1,4,2,5] => 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [4,1,3,2,5] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [2,3,1,4,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [2,4,1,3,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [3,2,1,4,5] => 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [4,2,1,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [3,4,1,2,5] => 2
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [4,3,1,2,5] => 2
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St001556
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,2]]
=> [1,2] => [1,2] => 0
[2,1] => [[1],[2]]
=> [2,1] => [2,1] => 0
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [2,3,1] => 0
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => [2,3,1] => 0
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 0
[1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 0
[1,4,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => 0
[2,3,4,1] => [[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 0
[2,4,3,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[3,1,4,2] => [[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[4,1,3,2] => [[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0
[4,2,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => 1
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 0
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 0
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => 0
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => 0
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 2
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 2
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001565
Mp00252: Permutations —restriction⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001565: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001565: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [2,1] => 0
[1,3,2] => [1,2] => [1,2] => [2,1] => 0
[2,1,3] => [2,1] => [2,1] => [1,2] => 0
[2,3,1] => [2,1] => [2,1] => [1,2] => 0
[3,1,2] => [1,2] => [1,2] => [2,1] => 0
[3,2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3,4] => [1,2,3] => [1,3,2] => [3,2,1] => 1
[1,2,4,3] => [1,2,3] => [1,3,2] => [3,2,1] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[1,4,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,1,4,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1,4] => [2,3,1] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[2,4,3,1] => [2,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => [3,2,1] => [2,1,3] => 0
[3,2,4,1] => [3,2,1] => [3,2,1] => [2,1,3] => 0
[3,4,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 0
[4,1,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[4,2,3,1] => [2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 0
[1,2,3,4,5] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 2
[1,2,3,5,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 2
[1,2,4,3,5] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 2
[1,2,5,4,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 2
Description
The number of arithmetic progressions of length 2 in a permutation.
For a permutation of length $n$, this is the number of indices $1 \leq i < j < k \leq n$ such that $\pi(k) - \pi(j) = \pi(j) - \pi(i)$.
Matching statistic: St000619
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[2,1,3] => [1,3,2] => [1,3,2] => [3,1,2] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,5,3] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,5,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [4,5,1,2,3] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => [4,5,1,2,3] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => 3 = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => [3,1,2,5,4] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => [3,1,2,5,4] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 3 = 2 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000816
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
St000816: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,2]]
=> [[1,2]]
=> [2] => 1 = 0 + 1
[2,1] => [[1],[2]]
=> [[1,2]]
=> [2] => 1 = 0 + 1
[1,2,3] => [[1,2,3]]
=> [[1,2,3]]
=> [3] => 1 = 0 + 1
[1,3,2] => [[1,2],[3]]
=> [[1,2,3]]
=> [3] => 1 = 0 + 1
[2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => 1 = 0 + 1
[2,3,1] => [[1,3],[2]]
=> [[1,2],[3]]
=> [2,1] => 1 = 0 + 1
[3,1,2] => [[1,2],[3]]
=> [[1,2,3]]
=> [3] => 1 = 0 + 1
[3,2,1] => [[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1] => 1 = 0 + 1
[1,2,3,4] => [[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => 1 = 0 + 1
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => 1 = 0 + 1
[1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 0 + 1
[1,3,4,2] => [[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 0 + 1
[1,4,2,3] => [[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => 1 = 0 + 1
[1,4,3,2] => [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 0 + 1
[2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[2,1,4,3] => [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[2,3,1,4] => [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[2,3,4,1] => [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[2,4,1,3] => [[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[2,4,3,1] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[3,1,2,4] => [[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 0 + 1
[3,1,4,2] => [[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => 1 = 0 + 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,2] => 2 = 1 + 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,2] => 2 = 1 + 1
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => 1 = 0 + 1
[3,4,2,1] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,2] => 2 = 1 + 1
[4,1,2,3] => [[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => 1 = 0 + 1
[4,1,3,2] => [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 0 + 1
[4,2,1,3] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [2,2] => 2 = 1 + 1
[4,3,1,2] => [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1] => 1 = 0 + 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,2] => 2 = 1 + 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => 1 = 0 + 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => 1 = 0 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1] => 1 = 0 + 1
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1] => 1 = 0 + 1
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => 1 = 0 + 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1] => 1 = 0 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3 = 2 + 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3 = 2 + 1
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3 = 2 + 1
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3 = 2 + 1
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3 = 2 + 1
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [3,2] => 3 = 2 + 1
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1] => 1 = 0 + 1
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => 1 = 0 + 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,2] => 3 = 2 + 1
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,2] => 3 = 2 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => 1 = 0 + 1
[1,4,5,3,2] => [[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,2] => 3 = 2 + 1
Description
The number of standard composition tableaux of the composition.
See [1, Def. 4.2.6].
Apparently, the total number of tableaux of given size is the number of involutions.
Matching statistic: St001899
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1]
=> [1]
=> 1 = 0 + 1
[2,1] => [1] => [1]
=> [1]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[2,1,3] => [2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,3,1] => [2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,2] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[3,2,1] => [2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,3,4,1] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,2,1,4] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[3,4,1,2] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,4,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,3,1,2] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
Description
The total number of irreducible representations contained in the higher Lie character for an integer partition.
Matching statistic: St001900
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001900: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001900: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1]
=> [1]
=> 1 = 0 + 1
[2,1] => [1] => [1]
=> [1]
=> 1 = 0 + 1
[1,2,3] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[2,1,3] => [2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,3,1] => [2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,2] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[3,2,1] => [2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,3,4,1] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,4,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,2,4] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,2,1,4] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[3,2,4,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[3,4,1,2] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,4,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,3,1,2] => [3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 3 = 2 + 1
Description
The number of distinct irreducible representations contained in the higher Lie character for an integer partition.
Matching statistic: St000837
Mp00252: Permutations —restriction⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St000837: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => ? ∊ {0,0}
[2,1] => [1] => ? ∊ {0,0}
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 0
[2,3,4,1] => [2,3,1] => 0
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 0
[3,1,2,4] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 0
[4,3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => 2
[1,2,3,5,4] => [1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 2
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => 2
[1,5,2,4,3] => [1,2,4,3] => 2
Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
The following 249 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000744The length of the path to the largest entry in a standard Young tableau. St001644The dimension of a graph. St000741The Colin de Verdière graph invariant. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000456The monochromatic index of a connected graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000454The largest eigenvalue of a graph if it is integral. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 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. St000256The number of parts from which one can substract 2 and still get an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000706The product of the factorials of the multiplicities of an integer partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001964The interval resolution global dimension of a poset. St000284The Plancherel distribution on integer partitions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001613The binary logarithm of the size of the center of a lattice. 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. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001095The number of non-isomorphic posets with precisely one further covering relation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. 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. St001320The minimal number of occurrences of the path-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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001736The total number of cycles in a graph. St001272The number of graphs with the same degree sequence. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001656The monophonic position number of a graph. St000370The genus of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-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. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001857The number of edges in the reduced word graph of a signed permutation. St000068The number of minimal elements in a poset. St000905The number of different multiplicities of parts of an integer composition. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits 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. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001330The hat guessing number of a graph. St000936The number of even values of the symmetric group character corresponding to the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001845The number of join irreducibles minus the rank of a lattice. St001624The breadth of a lattice. St001490The number of connected components of a skew partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001651The Frankl number of a lattice. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000422The energy of a graph, if it is integral. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000914The sum of the values of the Möbius function of a poset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001642The Prague dimension of a graph. St000679The pruning number of an ordered tree. St001271The competition number of a graph. St001875The number of simple modules with projective dimension at most 1. 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. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000911The number of maximal antichains of maximal size in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000312The number of leaves in a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000379The number of Hamiltonian cycles in a graph. St000552The number of cut vertices of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001119The length of a shortest maximal path in a graph. St001281The normalized isoperimetric number of a graph. St001310The number of induced diamond graphs in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001826The maximal number of leaves on a vertex of a graph. St000069The number of maximal elements of a poset. St000553The number of blocks of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001282The number of graphs with the same chromatic polynomial. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001743The discrepancy of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000917The open packing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001277The degeneracy of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001358The largest degree of a regular subgraph of a graph. St001654The monophonic hull number of a graph. St001716The 1-improper chromatic number of a graph. St001792The arboricity of a graph. St000258The burning number of a graph. St000916The packing number of a graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St000264The girth of a graph, which is not a tree. St001108The 2-dynamic chromatic number of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph.
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