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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000156
Mapping
Mp00310: Permutations —toric promotion⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [3,2,1] => 2
[1,3,2] => [2,3,1] => 3
[2,1,3] => [3,1,2] => 1
[2,3,1] => [1,3,2] => 2
[3,1,2] => [2,1,3] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,2,3,1] => 3
[1,2,4,3] => [4,3,1,2] => 4
[1,3,2,4] => [2,4,1,3] => 3
[1,3,4,2] => [2,3,4,1] => 6
[1,4,2,3] => [3,4,1,2] => 3
[1,4,3,2] => [3,1,2,4] => 1
[2,1,3,4] => [4,1,2,3] => 1
[2,1,4,3] => [4,1,3,2] => 2
[2,3,1,4] => [1,4,2,3] => 2
[2,3,4,1] => [4,2,1,3] => 2
[2,4,1,3] => [4,3,2,1] => 5
[2,4,3,1] => [1,4,3,2] => 3
[3,1,2,4] => [2,4,3,1] => 4
[3,1,4,2] => [1,2,3,4] => 0
[3,2,1,4] => [1,2,4,3] => 3
[3,2,4,1] => [2,1,4,3] => 4
[3,4,1,2] => [2,3,1,4] => 3
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [3,4,2,1] => 4
[4,1,3,2] => [3,2,4,1] => 5
[4,2,1,3] => [1,3,4,2] => 5
[4,2,3,1] => [3,1,4,2] => 4
[4,3,1,2] => [3,2,1,4] => 2
[4,3,2,1] => [1,3,2,4] => 2
[1,2,3,4,5] => [5,2,3,4,1] => 4
[1,2,3,5,4] => [5,2,4,1,3] => 6
[1,2,4,3,5] => [5,3,1,2,4] => 4
[1,2,4,5,3] => [5,3,4,1,2] => 8
[1,2,5,3,4] => [5,4,1,2,3] => 4
[1,2,5,4,3] => [5,4,1,3,2] => 5
[1,3,2,4,5] => [2,5,1,3,4] => 3
[1,3,2,5,4] => [2,5,1,4,3] => 4
[1,3,4,2,5] => [2,3,5,1,4] => 6
[1,3,4,5,2] => [2,3,4,5,1] => 10
[1,3,5,2,4] => [2,4,5,1,3] => 6
[1,3,5,4,2] => [2,4,1,3,5] => 3
[1,4,2,3,5] => [3,5,1,2,4] => 3
[1,4,2,5,3] => [3,5,1,4,2] => 4
[1,4,3,2,5] => [3,1,2,5,4] => 5
[1,4,3,5,2] => [3,1,2,4,5] => 1
[1,4,5,2,3] => [3,4,5,1,2] => 6
Description
The Denert index of a permutation. It is defined as den(σ)=#{1≤l<k≤n:σ(k)<σ(l)≤k}+#{1≤l<k≤n:σ(l)≤k<σ(k)}+#{1≤l<k≤n:k<σ(k)<σ(l)} where n is the size of σ. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic (exc,den) is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, exc is the number of weak exceedences, see [[St000155]].
Matching statistic: St000794
Mapping
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000794: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [3,2,1] => [2,3,1] => 2
[1,3,2] => [2,3,1] => [3,2,1] => 3
[2,1,3] => [3,1,2] => [3,1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,2,3,1] => [2,3,4,1] => 3
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 4
[1,3,2,4] => [2,4,1,3] => [4,2,1,3] => 3
[1,3,4,2] => [2,3,4,1] => [4,3,2,1] => 6
[1,4,2,3] => [3,4,1,2] => [4,1,3,2] => 3
[1,4,3,2] => [3,1,2,4] => [3,1,2,4] => 1
[2,1,3,4] => [4,1,2,3] => [4,1,2,3] => 1
[2,1,4,3] => [4,1,3,2] => [3,4,1,2] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [4,2,1,3] => [2,4,1,3] => 2
[2,4,1,3] => [4,3,2,1] => [3,2,4,1] => 5
[2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 3
[3,1,2,4] => [2,4,3,1] => [3,4,2,1] => 4
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 3
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 4
[3,4,1,2] => [2,3,1,4] => [3,2,1,4] => 3
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [3,4,2,1] => [4,2,3,1] => 4
[4,1,3,2] => [3,2,4,1] => [2,4,3,1] => 5
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 5
[4,2,3,1] => [3,1,4,2] => [4,3,1,2] => 4
[4,3,1,2] => [3,2,1,4] => [2,3,1,4] => 2
[4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[1,2,3,4,5] => [5,2,3,4,1] => [2,3,4,5,1] => 4
[1,2,3,5,4] => [5,2,4,1,3] => [2,4,1,5,3] => 6
[1,2,4,3,5] => [5,3,1,2,4] => [3,1,5,2,4] => 4
[1,2,4,5,3] => [5,3,4,1,2] => [4,3,1,5,2] => 8
[1,2,5,3,4] => [5,4,1,2,3] => [4,1,5,2,3] => 4
[1,2,5,4,3] => [5,4,1,3,2] => [4,1,3,5,2] => 5
[1,3,2,4,5] => [2,5,1,3,4] => [5,2,1,3,4] => 3
[1,3,2,5,4] => [2,5,1,4,3] => [4,5,2,1,3] => 4
[1,3,4,2,5] => [2,3,5,1,4] => [5,3,2,1,4] => 6
[1,3,4,5,2] => [2,3,4,5,1] => [5,4,3,2,1] => 10
[1,3,5,2,4] => [2,4,5,1,3] => [5,2,1,4,3] => 6
[1,3,5,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,4,2,3,5] => [3,5,1,2,4] => [5,1,3,2,4] => 3
[1,4,2,5,3] => [3,5,1,4,2] => [4,5,1,3,2] => 4
[1,4,3,2,5] => [3,1,2,5,4] => [3,1,2,5,4] => 5
[1,4,3,5,2] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,4,5,2,3] => [3,4,5,1,2] => [5,1,4,3,2] => 6
[1,4,5,3,2] => [3,4,1,2,5] => [4,1,3,2,5] => 3
Description
The mak of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns (231_), (32_1), (132_), (21_), where matches of the underlined letters must be adjacent.