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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000356
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(load all 6 compositions to match this statistic)
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[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] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[5,1,2,3,4] => 0
[5,1,2,4,3] => 1
[5,1,3,2,4] => 1
[5,1,3,4,2] => 1
[5,1,4,2,3] => 2
[5,1,4,3,2] => 2
[5,2,1,3,4] => 0
[5,2,1,4,3] => 1
[5,2,3,1,4] => 0
[5,2,3,4,1] => 0
[5,2,4,1,3] => 1
[5,2,4,3,1] => 1
[5,3,1,2,4] => 0
[5,3,1,4,2] => 1
[5,3,2,1,4] => 0
[5,3,2,4,1] => 0
[5,3,4,1,2] => 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000358
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 1
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2] => [4,1,2,3] => 2
[2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,2,4,1] => 0
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[5,1,2,3,4] => [1,5,4,3,2] => 0
[5,1,2,4,3] => [1,5,4,2,3] => 1
[5,1,3,2,4] => [1,5,3,4,2] => 1
[5,1,3,4,2] => [1,5,3,2,4] => 1
[5,1,4,2,3] => [1,5,2,4,3] => 2
[5,1,4,3,2] => [1,5,2,3,4] => 2
[5,2,1,3,4] => [1,4,5,3,2] => 0
[5,2,1,4,3] => [1,4,5,2,3] => 1
[5,2,3,1,4] => [1,4,3,5,2] => 0
[5,2,3,4,1] => [1,4,3,2,5] => 0
[5,2,4,1,3] => [1,4,2,5,3] => 1
[5,2,4,3,1] => [1,4,2,3,5] => 1
[5,3,1,2,4] => [1,3,5,4,2] => 0
[5,3,1,4,2] => [1,3,5,2,4] => 1
[5,3,2,1,4] => [1,3,4,5,2] => 0
[5,3,2,4,1] => [1,3,4,2,5] => 0
[5,3,4,1,2] => [1,3,2,5,4] => 0
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001683
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001683: 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] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => 0
[3,1,2] => [2,3,1] => 0
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => [1,3,4,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,1,2,4] => 0
[2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [4,1,3,2] => 1
[3,1,2,4] => [2,3,1,4] => 0
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => 0
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => 0
[4,1,3,2] => [2,4,3,1] => 1
[4,2,1,3] => [3,2,4,1] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [3,4,2,1] => 0
[4,3,2,1] => [4,3,2,1] => 0
[5,1,2,3,4] => [2,3,4,5,1] => 0
[5,1,2,4,3] => [2,3,5,4,1] => 1
[5,1,3,2,4] => [2,4,3,5,1] => 1
[5,1,3,4,2] => [2,5,3,4,1] => 1
[5,1,4,2,3] => [2,4,5,3,1] => 2
[5,1,4,3,2] => [2,5,4,3,1] => 2
[5,2,1,3,4] => [3,2,4,5,1] => 0
[5,2,1,4,3] => [3,2,5,4,1] => 1
[5,2,3,1,4] => [4,2,3,5,1] => 0
[5,2,3,4,1] => [5,2,3,4,1] => 0
[5,2,4,1,3] => [4,2,5,3,1] => 1
[5,2,4,3,1] => [5,2,4,3,1] => 1
[5,3,1,2,4] => [3,4,2,5,1] => 0
[5,3,1,4,2] => [3,5,2,4,1] => 1
[5,3,2,1,4] => [4,3,2,5,1] => 0
[5,3,2,4,1] => [5,3,2,4,1] => 0
[5,3,4,1,2] => [4,5,2,3,1] => 0
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St000223
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of nestings in the permutation.
Matching statistic: St000371
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => 1
[5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => 1
[5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => 0
[5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => 0
[5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St001687
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [2,1,3] => 1
[2,1,3] => [3,1,2] => [1,3,2] => 0
[2,3,1] => [1,3,2] => [3,1,2] => 0
[3,1,2] => [2,1,3] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 1
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 1
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 2
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 2
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 0
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 1
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 0
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => 1
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 0
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => 0
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 1
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 0
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[5,1,2,4,3] => [3,4,2,1,5] => [3,2,4,5,1] => 1
[5,1,3,2,4] => [4,2,3,1,5] => [2,4,3,5,1] => 1
[5,1,3,4,2] => [2,4,3,1,5] => [4,2,3,5,1] => 1
[5,1,4,2,3] => [3,2,4,1,5] => [3,4,2,5,1] => 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,3,2,5,1] => 2
[5,2,1,3,4] => [4,3,1,2,5] => [2,3,5,4,1] => 0
[5,2,1,4,3] => [3,4,1,2,5] => [3,2,5,4,1] => 1
[5,2,3,1,4] => [4,1,3,2,5] => [2,5,3,4,1] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[5,2,4,1,3] => [3,1,4,2,5] => [3,5,2,4,1] => 1
[5,2,4,3,1] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,5,3,1] => 0
[5,3,1,4,2] => [2,4,1,3,5] => [4,2,5,3,1] => 1
[5,3,2,1,4] => [4,1,2,3,5] => [2,5,4,3,1] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [5,2,4,3,1] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [4,5,2,3,1] => 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St000039
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [2,4,3,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => [4,3,2,1] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [3,4,2,1] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => [4,2,3,1,5] => 0
[5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => [3,2,4,1,5] => 1
[5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => [4,3,1,2,5] => 1
[5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => [2,4,3,1,5] => 1
[5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => [3,4,1,2,5] => 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 2
[5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => [4,1,3,2,5] => 0
[5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => [3,1,4,2,5] => 1
[5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 0
[5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 0
[5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000372
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [1,3,2] => 0
[1,3,2] => [2,3,1] => [3,2,1] => [1,2,3] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [2,1,3] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [3,1,2] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [1,4,3,2] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [1,3,4,2] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [1,2,4,3] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [1,4,2,3] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [2,1,3,4] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,1,2,4] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => [5,1,4,3,2] => 0
[5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => [5,1,3,4,2] => 1
[5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => [5,1,2,4,3] => 1
[5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => [5,1,4,2,3] => 1
[5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => [5,1,3,2,4] => 2
[5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => [5,2,4,1,3] => 0
[5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => [5,2,3,1,4] => 1
[5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => [5,2,1,4,3] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 0
[5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => [5,2,1,3,4] => 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => [5,3,1,4,2] => 0
[5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => [5,3,1,2,4] => 1
[5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => [5,3,2,1,4] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 0
Description
The number of mid points of increasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) < \pi(j) < \pi(k)$.
The generating function is given by [1].
Matching statistic: St001882
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => [1,3,4,2] => 1
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[5,1,2,4,3] => [2,3,5,4,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[5,1,3,2,4] => [2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[5,1,3,4,2] => [2,5,3,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[5,1,4,2,3] => [2,4,5,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[5,1,4,3,2] => [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[5,2,1,3,4] => [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[5,2,1,4,3] => [3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[5,2,3,1,4] => [4,2,3,5,1] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[5,2,3,4,1] => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,2,4,1,3] => [4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[5,3,1,2,4] => [3,4,2,5,1] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[5,3,1,4,2] => [3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[5,3,2,1,4] => [4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[5,3,2,4,1] => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,1,2] => [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St001866
Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 0
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[2,3,1,4] => [4,1,3,2] => [3,4,1,2] => [3,4,1,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[5,1,2,3,4] => [4,3,2,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[5,1,2,4,3] => [3,4,2,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[5,1,3,2,4] => [4,2,3,1,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[5,1,3,4,2] => [2,4,3,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[5,1,4,2,3] => [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2
[5,2,1,3,4] => [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 0
[5,2,1,4,3] => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[5,2,3,1,4] => [4,1,3,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[5,2,4,1,3] => [3,1,4,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[5,3,1,2,4] => [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[5,3,1,4,2] => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[5,4,1,2,3] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[5,4,1,3,2] => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
* $-i < -j < -\pi(j) < -\pi(i)$, or
* $-i < j \leq \pi(j) < -\pi(i)$, or
* $i < j \leq \pi(j) < \pi(i)$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001857The number of edges in the reduced word graph of a signed permutation. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000709The number of occurrences of 14-2-3 or 14-3-2. 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. St001868The number of alignments of type NE of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000264The girth of a graph, which is not a tree. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001964The interval resolution global dimension of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000181The number of connected components of the Hasse diagram for the poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001890The maximum magnitude of the Möbius function of a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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