Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000538
Mapping
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%
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
Mapping
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%
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: St000837
Mapping
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1] => [1] => ? = 0
[2,1] => [1,2] => [1] => [1] => ? = 0
[1,2,3] => [3,2,1] => [2,1] => [2,1] => 0
[1,3,2] => [3,1,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,3,1] => [2,1] => [2,1] => 0
[2,3,1] => [2,1,3] => [2,1] => [2,1] => 0
[3,1,2] => [1,3,2] => [1,2] => [1,2] => 0
[3,2,1] => [1,2,3] => [1,2] => [1,2] => 0
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => [3,1,2] => [2,3,1] => 0
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [3,1,2] => 0
[1,3,4,2] => [4,2,1,3] => [2,1,3] => [2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => [1,3,2] => [1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => [1,2,3] => [1,2,3] => 1
[2,1,3,4] => [3,4,2,1] => [3,2,1] => [3,2,1] => 0
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [2,3,1] => 0
[2,3,1,4] => [3,2,4,1] => [3,2,1] => [3,2,1] => 0
[2,3,4,1] => [3,2,1,4] => [3,2,1] => [3,2,1] => 0
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [2,3,1] => 0
[2,4,3,1] => [3,1,2,4] => [3,1,2] => [2,3,1] => 0
[3,1,2,4] => [2,4,3,1] => [2,3,1] => [3,1,2] => 0
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1,4] => [2,3,4,1] => [2,3,1] => [3,1,2] => 0
[3,2,4,1] => [2,3,1,4] => [2,3,1] => [3,1,2] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[3,4,2,1] => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[4,1,2,3] => [1,4,3,2] => [1,3,2] => [1,3,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,2,3] => [1,2,3] => 1
[4,2,1,3] => [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,3] => [1,2,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 0
[1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
[1,2,4,5,3] => [5,4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 0
[1,3,2,5,4] => [5,3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
[1,3,4,2,5] => [5,3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 0
[1,3,5,4,2] => [5,3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,4,2,3,5] => [5,2,4,3,1] => [2,4,3,1] => [4,1,3,2] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,4,5,2,3] => [5,2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,4,5,3,2] => [5,2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,5,2,3,4] => [5,1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
Description
The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.