- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000371
(load all 5 compositions to match this statistic)

Mapping

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000371: 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] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 2
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 0
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [3,4,2,1] => 1
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [4,3,1,2] => 1
[3,1,4,2] => [4,3,1,2] => [4,1,3,2] => [2,4,3,1] => 1
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 1
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [2,4,1,3] => 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 0
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [1,3,4,2] => 0
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [2,3,1,4] => 0
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => [4,1,2,5,3] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [5,1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => [5,1,4,3,2] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => [2,1,5,3,4] => 0
[1,3,5,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => [4,1,5,3,2] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => [5,1,4,2,3] => 1
[1,4,2,5,3] => [1,5,4,2,3] => [2,5,1,4,3] => [3,1,5,4,2] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,4,1,5,3] => [3,1,5,2,4] => 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 3 compositions to match this statistic)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 56%

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] => [2,3,1] => [3,1,2] => [2,3,1] => 0
[1,3,2] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [2,1,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [2,1,3] => [2,1,3] => [1,3,2] => 0
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 0
[1,2,4,3] => [2,4,3,1] => [4,1,3,2] => [2,4,3,1] => 0
[1,3,2,4] => [3,2,4,1] => [4,2,1,3] => [3,2,4,1] => 1
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => [3,4,2,1] => 0
[1,4,2,3] => [3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 0
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [2,1,3,4] => 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 1
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,4,2,3] => [1,3,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,2,4,3] => 0
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [3,1,2,4] => 1
[3,1,4,2] => [4,1,3,2] => [2,4,3,1] => [3,1,4,2] => 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [1,3,2,4] => 1
[3,4,1,2] => [4,1,2,3] => [2,3,4,1] => [3,4,1,2] => 0
[3,4,2,1] => [3,1,2,4] => [2,3,1,4] => [1,3,4,2] => 0
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [4,1,2,3] => 0
[4,1,3,2] => [4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 0
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [1,4,2,3] => 0
[4,3,1,2] => [4,2,1,3] => [3,2,4,1] => [4,3,1,2] => 0
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => [1,4,3,2] => 0
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => [2,3,5,4,1] => 0
[1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => [2,4,3,5,1] => 1
[1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => [2,4,5,3,1] => 0
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,4,2,3] => [2,5,3,4,1] => 0
[1,2,5,4,3] => [2,5,4,3,1] => [5,1,4,3,2] => [2,5,4,3,1] => 0
[1,3,2,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[1,3,2,5,4] => [3,2,5,4,1] => [5,2,1,4,3] => [3,2,5,4,1] => 1
[1,3,4,2,5] => [4,2,3,5,1] => [5,2,3,1,4] => [3,4,2,5,1] => 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [3,4,5,2,1] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [3,5,2,4,1] => 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [3,5,4,2,1] => 0
[1,4,2,3,5] => [3,4,2,5,1] => [5,3,1,2,4] => [4,2,3,5,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [4,2,5,3,1] => 1
[1,4,3,2,5] => [4,3,2,5,1] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [4,3,5,2,1] => 1
[1,4,5,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => [4,5,2,3,1] => 0
[1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => [2,3,4,5,7,6,1] => ? = 0
[1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [2,3,4,6,7,5,1] => ? = 0
[1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => [2,3,4,7,5,6,1] => ? = 0
[1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [2,3,4,7,6,5,1] => ? = 0
[1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => ? = 0
[1,2,3,6,7,5,4] => [2,3,7,6,4,5,1] => [7,1,2,5,6,4,3] => [2,3,6,7,5,4,1] => ? = 0
[1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [2,3,7,4,5,6,1] => ? = 0
[1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => [2,3,7,4,6,5,1] => ? = 0
[1,2,3,7,6,4,5] => [2,3,6,7,5,4,1] => [7,1,2,6,5,3,4] => [2,3,7,6,4,5,1] => ? = 0
[1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [2,3,7,6,5,4,1] => ? = 0
[1,2,6,7,5,4,3] => [2,7,6,5,3,4,1] => [7,1,5,6,4,3,2] => [2,6,7,5,4,3,1] => ? = 0
[1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => [2,7,3,4,5,6,1] => ? = 0
[1,2,7,6,3,4,5] => [2,5,6,7,4,3,1] => [7,1,6,5,2,3,4] => [2,7,6,3,4,5,1] => ? = 0
[1,2,7,6,5,4,3] => [2,7,6,5,4,3,1] => [7,1,6,5,4,3,2] => [2,7,6,5,4,3,1] => ? = 0
[1,3,4,5,6,7,2] => [7,2,3,4,5,6,1] => [7,2,3,4,5,6,1] => [3,4,5,6,7,2,1] => ? = 0
[1,3,4,5,7,6,2] => [7,2,3,4,6,5,1] => [7,2,3,4,6,5,1] => [3,4,5,7,6,2,1] => ? = 0
[1,4,5,6,7,3,2] => [7,6,2,3,4,5,1] => [7,3,4,5,6,2,1] => [4,5,6,7,3,2,1] => ? = 0
[1,5,4,6,3,7,2] => [7,5,3,2,4,6,1] => [7,4,3,5,2,6,1] => [5,4,6,3,7,2,1] => ? = 3
[1,5,6,7,4,3,2] => [7,6,5,2,3,4,1] => [7,4,5,6,3,2,1] => [5,6,7,4,3,2,1] => ? = 0
[1,5,7,6,2,3,4] => [5,6,7,2,4,3,1] => [7,4,6,5,1,2,3] => [5,7,6,2,3,4,1] => ? = 0
[1,7,2,3,4,6,5] => [3,4,5,7,6,2,1] => [7,6,1,2,3,5,4] => [7,2,3,4,6,5,1] => ? = 0
[1,7,6,2,3,4,5] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => [7,6,2,3,4,5,1] => ? = 0
[1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => [2,1,3,4,5,6,7] => ? = 1
[2,1,7,6,5,4,3] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [2,1,7,6,5,4,3] => ? = 1
[2,3,1,4,5,6,7] => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [2,3,1,4,5,6,7] => ? = 2
[2,3,1,7,6,5,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [2,3,1,7,6,5,4] => ? = 2
[2,3,7,1,6,5,4] => [1,2,7,6,5,3,4] => [1,2,6,7,5,4,3] => [2,3,7,1,6,5,4] => ? = 2
[2,4,1,3,5,6,7] => [1,4,2,5,6,7,3] => [1,3,7,2,4,5,6] => [2,4,1,3,5,6,7] => ? = 2
[2,4,3,1,5,6,7] => [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => [2,4,3,1,5,6,7] => ? = 3
[2,4,5,3,1,7,6] => [1,4,2,3,7,6,5] => [1,3,4,2,7,6,5] => [2,4,5,3,1,7,6] => ? = 4
[2,7,1,6,5,4,3] => [1,7,6,5,4,2,3] => [1,6,7,5,4,3,2] => [2,7,1,6,5,4,3] => ? = 1
[2,7,3,1,6,5,4] => [1,3,7,6,5,2,4] => [1,6,2,7,5,4,3] => [2,7,3,1,6,5,4] => ? = 2
[2,7,4,3,1,6,5] => [1,4,3,7,6,2,5] => [1,6,3,2,7,5,4] => [2,7,4,3,1,6,5] => ? = 3
[2,7,6,3,1,4,5] => [1,4,6,7,3,2,5] => [1,6,5,2,7,3,4] => [2,7,6,3,1,4,5] => ? = 2
[3,1,2,4,5,6,7] => [3,1,4,5,6,7,2] => [2,7,1,3,4,5,6] => [3,1,2,4,5,6,7] => ? = 1
[3,2,1,4,5,6,7] => [2,1,4,5,6,7,3] => [2,1,7,3,4,5,6] => [3,2,1,4,5,6,7] => ? = 2
[3,2,4,1,6,7,5] => [2,1,3,7,5,6,4] => [2,1,3,7,5,6,4] => [3,2,4,1,6,7,5] => ? = 3
[3,2,4,1,7,6,5] => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => [3,2,4,1,7,6,5] => ? = 3
[3,2,5,4,1,7,6] => [2,1,4,3,7,6,5] => [2,1,4,3,7,6,5] => [3,2,5,4,1,7,6] => ? = 4
[3,2,7,4,1,5,6] => [2,1,4,6,7,3,5] => [2,1,6,3,7,4,5] => [3,2,7,4,1,5,6] => ? = 3
[3,2,7,6,4,1,5] => [2,1,5,7,4,3,6] => [2,1,6,5,3,7,4] => [3,2,7,6,4,1,5] => ? = 3
[3,4,2,5,1,7,6] => [3,1,2,4,7,6,5] => [2,3,1,4,7,6,5] => [3,4,2,5,1,7,6] => ? = 4
[3,4,2,7,6,1,5] => [3,1,2,7,5,4,6] => [2,3,1,6,5,7,4] => [3,4,2,7,6,1,5] => ? = 3
[3,4,5,1,6,2,7] => [6,1,2,3,5,7,4] => [2,3,4,7,5,1,6] => [3,4,5,1,6,2,7] => ? = 4
[3,4,5,6,7,1,2] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => ? = 0
[3,4,5,7,6,1,2] => [7,1,2,3,5,4,6] => [2,3,4,6,5,7,1] => [3,4,5,7,6,1,2] => ? = 0
[3,4,6,1,5,2,7] => [6,1,2,5,3,7,4] => [2,3,5,7,4,1,6] => [3,4,6,1,5,2,7] => ? = 4
[3,4,7,1,5,2,6] => [6,1,2,5,7,3,4] => [2,3,6,7,4,1,5] => [3,4,7,1,5,2,6] => ? = 3

Description

The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.

Matching statistic: St001683
(load all 2 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 67%

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] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [3,1,2] => [2,1,3] => 0
[2,1,3] => [3,1,2] => [2,3,1] => [1,3,2] => 1
[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] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [2,1,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [2,3,1,4] => 0
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [3,1,2,4] => 0
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 0
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [1,2,4,3] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [1,3,4,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [1,4,2,3] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 2
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [2,4,3,1] => 1
[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] => [3,2,1,4] => [4,1,2,3] => 0
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 0
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [4,1,3,2] => 1
[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
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [2,1,3,4,5] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [1,3,2,4,5] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [2,3,1,4,5] => 0
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [3,1,2,4,5] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [3,2,1,4,5] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [1,2,4,3,5] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [2,1,4,3,5] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,3,4,2,5] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [2,3,4,1,5] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [3,1,4,2,5] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [3,2,4,1,5] => 0
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [2,4,1,3,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [2,4,3,1,5] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [3,4,1,2,5] => 0
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => ? = 0
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [7,6,5,4,1,3,2] => [2,3,1,4,5,6,7] => ? = 0
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => [3,1,2,4,5,6,7] => ? = 0
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => ? = 0
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [7,6,5,2,1,4,3] => [3,4,1,2,5,6,7] => ? = 0
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [7,6,5,1,2,4,3] => [3,4,2,1,5,6,7] => ? = 0
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => [4,1,2,3,5,6,7] => ? = 0
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => [4,2,1,3,5,6,7] => ? = 0
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => [4,3,1,2,5,6,7] => ? = 0
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => ? = 0
[1,2,6,7,5,4,3] => [3,4,5,7,6,2,1] => [7,6,1,2,3,5,4] => [4,5,3,2,1,6,7] => ? = 0
[1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => [5,1,2,3,4,6,7] => ? = 0
[1,2,7,6,3,4,5] => [5,4,3,6,7,2,1] => [7,6,3,2,1,4,5] => [5,4,1,2,3,6,7] => ? = 0
[1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => ? = 0
[1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => [7,1,6,5,4,3,2] => [2,3,4,5,6,1,7] => ? = 0
[1,3,4,5,7,6,2] => [2,6,7,5,4,3,1] => [7,1,6,5,4,2,3] => [3,2,4,5,6,1,7] => ? = 0
[1,4,5,6,7,3,2] => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [3,4,5,6,2,1,7] => ? = 0
[1,5,4,6,3,7,2] => [2,7,3,6,4,5,1] => [7,1,3,5,6,4,2] => [2,4,6,5,3,1,7] => ? = 3
[1,5,6,7,4,3,2] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [4,5,6,3,2,1,7] => ? = 0
[1,5,7,6,2,3,4] => [4,3,2,6,7,5,1] => [7,3,2,1,6,4,5] => [5,4,6,1,2,3,7] => ? = 0
[1,7,2,3,4,6,5] => [5,6,4,3,2,7,1] => [7,5,4,3,1,2,6] => [6,2,1,3,4,5,7] => ? = 0
[1,7,6,2,3,4,5] => [5,4,3,2,6,7,1] => [7,4,3,2,1,5,6] => [6,5,1,2,3,4,7] => ? = 0
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => ? = 0
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => ? = 1
[2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => [5,7,6,1,2,3,4] => [4,3,2,1,6,7,5] => ? = 2
[2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[2,3,4,6,7,5,1] => [1,5,7,6,4,3,2] => [1,7,6,5,2,4,3] => [3,4,2,5,6,7,1] => ? = 0
[2,3,4,7,5,6,1] => [1,6,5,7,4,3,2] => [1,7,6,5,3,2,4] => [4,2,3,5,6,7,1] => ? = 0
[2,3,5,6,7,4,1] => [1,4,7,6,5,3,2] => [1,7,6,2,5,4,3] => [3,4,5,2,6,7,1] => ? = 0
[2,3,5,7,6,4,1] => [1,4,6,7,5,3,2] => [1,7,6,2,5,3,4] => [4,3,5,2,6,7,1] => ? = 0
[2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => [1,7,6,2,4,5,3] => [3,5,4,2,6,7,1] => ? = 1
[2,3,6,7,5,4,1] => [1,4,5,7,6,3,2] => [1,7,6,2,3,5,4] => [4,5,3,2,6,7,1] => ? = 0
[2,3,7,1,6,5,4] => [4,5,6,1,7,3,2] => [4,7,6,1,2,3,5] => [5,3,2,1,6,7,4] => ? = 2
[2,3,7,4,5,6,1] => [1,6,5,4,7,3,2] => [1,7,6,4,3,2,5] => [5,2,3,4,6,7,1] => ? = 0
[2,3,7,5,6,4,1] => [1,4,6,5,7,3,2] => [1,7,6,2,4,3,5] => [5,3,4,2,6,7,1] => ? = 0
[2,3,7,6,5,4,1] => [1,4,5,6,7,3,2] => [1,7,6,2,3,4,5] => [5,4,3,2,6,7,1] => ? = 0
[2,4,5,3,1,7,6] => [6,7,1,3,5,4,2] => [3,7,4,6,5,1,2] => [2,1,5,6,4,7,3] => ? = 4
[2,4,5,6,7,3,1] => [1,3,7,6,5,4,2] => [1,7,2,6,5,4,3] => [3,4,5,6,2,7,1] => ? = 0
[2,4,5,7,6,3,1] => [1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => [4,3,5,6,2,7,1] => ? = 0
[2,4,6,5,7,3,1] => [1,3,7,5,6,4,2] => [1,7,2,6,4,5,3] => [3,5,4,6,2,7,1] => ? = 1
[2,4,6,7,5,3,1] => [1,3,5,7,6,4,2] => [1,7,2,6,3,5,4] => [4,5,3,6,2,7,1] => ? = 0
[2,4,7,5,6,3,1] => [1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => [5,3,4,6,2,7,1] => ? = 0
[2,4,7,6,5,3,1] => [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [5,4,3,6,2,7,1] => ? = 0
[2,5,3,6,7,4,1] => [1,4,7,6,3,5,2] => [1,7,5,2,6,4,3] => [3,4,6,2,5,7,1] => ? = 1
[2,5,3,7,6,4,1] => [1,4,6,7,3,5,2] => [1,7,5,2,6,3,4] => [4,3,6,2,5,7,1] => ? = 1
[2,5,4,6,7,3,1] => [1,3,7,6,4,5,2] => [1,7,2,5,6,4,3] => [3,4,6,5,2,7,1] => ? = 1
[2,5,4,7,6,3,1] => [1,3,6,7,4,5,2] => [1,7,2,5,6,3,4] => [4,3,6,5,2,7,1] => ? = 1
[2,5,6,3,7,4,1] => [1,4,7,3,6,5,2] => [1,7,4,2,6,5,3] => [3,5,6,2,4,7,1] => ? = 2
[2,5,6,4,7,3,1] => [1,3,7,4,6,5,2] => [1,7,2,4,6,5,3] => [3,5,6,4,2,7,1] => ? = 2
[2,5,6,7,3,4,1] => [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => [4,5,6,2,3,7,1] => ? = 0

Description

The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.

Matching statistic: St000372
(load all 4 compositions to match this statistic)

Mapping

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,3,1] => [1,2,3] => 1
[2,3,1] => [3,2,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [3,1,2] => [1,3,2] => [2,1,3] => 0
[3,2,1] => [2,3,1] => [2,1,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 0
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 0
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 0
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => [1,4,2,3] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,1,2,3] => 1
[2,3,1,4] => [3,2,1,4] => [2,3,4,1] => [1,2,3,4] => 2
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => [2,4,3,1] => 0
[2,4,1,3] => [4,2,1,3] => [1,3,4,2] => [2,1,3,4] => 1
[2,4,3,1] => [3,2,4,1] => [2,3,1,4] => [4,2,3,1] => 0
[3,1,2,4] => [3,1,2,4] => [2,4,3,1] => [1,2,4,3] => 1
[3,1,4,2] => [4,3,1,2] => [1,2,4,3] => [2,3,1,4] => 1
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => [1,3,2,4] => 2
[3,2,4,1] => [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 1
[3,4,1,2] => [4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 0
[3,4,2,1] => [2,4,3,1] => [3,1,2,4] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 0
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => [3,2,1,4] => 0
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,2,4] => 1
[4,2,3,1] => [3,4,2,1] => [2,1,3,4] => [3,2,4,1] => 0
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 0
[4,3,2,1] => [2,3,4,1] => [3,2,1,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => [4,5,1,3,2] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [4,1,5,3,2] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [1,5,3,4,2] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [5,1,3,4,2] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,3,4,5,2] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [5,1,3,2,4] => [3,5,4,1,2] => 0
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => [3,1,4,5,2] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [5,2,3,1,4] => [5,3,4,1,2] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [1,3,5,4,2] => 1
[1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => [3,4,1,5,2] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [1,4,3,5,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [3,4,5,1,2] => 1
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => [3,5,1,4,2] => 0
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [7,1,6,5,4,3,2] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [6,7,1,5,4,3,2] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 0
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,1,5,4,3,2] => ? = 0
[1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [5,7,1,6,4,3,2] => ? = 0
[1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [6,7,5,1,4,3,2] => ? = 0
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [5,1,7,6,4,3,2] => ? = 0
[1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [6,5,1,7,4,3,2] => ? = 0
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [7,5,1,6,4,3,2] => ? = 0
[1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [7,6,5,1,4,3,2] => ? = 0
[1,2,6,7,5,4,3] => [1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [6,7,5,4,1,3,2] => ? = 0
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => [4,1,7,6,5,3,2] => ? = 0
[1,2,7,6,3,4,5] => [1,2,6,3,4,7,5] => [7,6,2,5,4,1,3] => [7,4,1,6,5,3,2] => ? = 0
[1,2,7,6,5,4,3] => [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [7,6,5,4,1,3,2] => ? = 0
[1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => [7,1,5,4,3,2,6] => [3,7,6,5,4,1,2] => ? = 0
[1,3,4,5,7,6,2] => [1,6,3,4,5,7,2] => [7,2,5,4,3,1,6] => [7,3,6,5,4,1,2] => ? = 0
[1,4,5,6,7,3,2] => [1,3,7,4,5,6,2] => [7,5,1,4,3,2,6] => [4,7,6,5,3,1,2] => ? = 0
[1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [7,1,2,3,4,5,6] => [3,4,5,6,7,1,2] => ? = 3
[1,5,6,7,4,3,2] => [1,3,4,7,5,6,2] => [7,5,4,1,3,2,6] => [5,7,6,4,3,1,2] => ? = 0
[1,5,7,6,2,3,4] => [1,6,2,3,5,7,4] => [7,2,6,5,3,1,4] => [7,3,6,1,5,4,2] => ? = 0
[1,7,2,3,4,6,5] => [1,6,2,3,7,4,5] => [7,2,6,5,1,4,3] => [6,3,1,7,5,4,2] => ? = 0
[1,7,6,2,3,4,5] => [1,6,2,3,4,7,5] => [7,2,6,5,4,1,3] => [7,3,1,6,5,4,2] => ? = 0
[1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => [7,5,4,3,2,1,6] => [7,6,5,4,3,1,2] => ? = 0
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => [1,7,6,5,4,2,3] => ? = 1
[2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => [6,7,4,3,2,1,5] => [7,6,5,4,1,2,3] => ? = 1
[2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => ? = 2
[2,3,1,7,6,5,4] => [3,2,1,5,6,7,4] => [5,6,7,3,2,1,4] => [7,6,5,1,2,3,4] => ? = 2
[2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => [2,7,6,5,4,3,1] => ? = 0
[2,3,4,6,7,5,1] => [5,2,3,4,7,6,1] => [3,6,5,4,1,2,7] => [6,7,2,5,4,3,1] => ? = 0
[2,3,4,7,5,6,1] => [6,2,3,4,7,5,1] => [2,6,5,4,1,3,7] => [6,2,7,5,4,3,1] => ? = 0
[2,3,5,6,7,4,1] => [4,2,3,7,5,6,1] => [4,6,5,1,3,2,7] => [5,7,6,2,4,3,1] => ? = 0
[2,3,5,7,6,4,1] => [4,2,3,6,5,7,1] => [4,6,5,2,3,1,7] => [7,5,6,2,4,3,1] => ? = 0
[2,3,6,5,7,4,1] => [4,2,3,7,6,5,1] => [4,6,5,1,2,3,7] => [5,6,7,2,4,3,1] => ? = 1
[2,3,6,7,5,4,1] => [4,2,3,5,7,6,1] => [4,6,5,3,1,2,7] => [6,7,5,2,4,3,1] => ? = 0
[2,3,7,1,6,5,4] => [5,2,3,6,7,1,4] => [3,6,5,2,1,7,4] => [6,5,2,1,4,3,7] => ? = 2
[2,3,7,4,5,6,1] => [6,2,3,7,4,5,1] => [2,6,5,1,4,3,7] => [5,2,7,6,4,3,1] => ? = 0
[2,3,7,5,6,4,1] => [4,2,3,6,7,5,1] => [4,6,5,2,1,3,7] => [6,5,7,2,4,3,1] => ? = 0
[2,3,7,6,5,4,1] => [4,2,3,5,6,7,1] => [4,6,5,3,2,1,7] => [7,6,5,2,4,3,1] => ? = 0
[2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => [4,6,7,5,3,2,1] => [1,7,6,2,5,3,4] => ? = 2
[2,4,3,1,5,6,7] => [3,2,4,1,5,6,7] => [5,6,4,7,3,2,1] => [1,7,6,4,2,3,5] => ? = 3
[2,4,5,3,1,7,6] => [3,2,5,4,1,7,6] => [5,6,3,4,7,1,2] => [7,1,4,5,2,3,6] => ? = 4
[2,4,5,6,7,3,1] => [3,2,7,4,5,6,1] => [5,6,1,4,3,2,7] => [4,7,6,5,2,3,1] => ? = 0
[2,4,5,7,6,3,1] => [3,2,6,4,5,7,1] => [5,6,2,4,3,1,7] => [7,4,6,5,2,3,1] => ? = 0
[2,4,6,5,7,3,1] => [3,2,7,4,6,5,1] => [5,6,1,4,2,3,7] => [4,6,7,5,2,3,1] => ? = 1
[2,4,6,7,5,3,1] => [3,2,5,4,7,6,1] => [5,6,3,4,1,2,7] => [6,7,4,5,2,3,1] => ? = 0
[2,4,7,5,6,3,1] => [3,2,6,4,7,5,1] => [5,6,2,4,1,3,7] => [6,4,7,5,2,3,1] => ? = 0
[2,4,7,6,5,3,1] => [3,2,5,4,6,7,1] => [5,6,3,4,2,1,7] => [7,6,4,5,2,3,1] => ? = 0
[2,5,3,6,7,4,1] => [4,2,7,5,3,6,1] => [4,6,1,3,5,2,7] => [4,7,5,2,6,3,1] => ? = 1
[2,5,3,7,6,4,1] => [4,2,6,5,3,7,1] => [4,6,2,3,5,1,7] => [7,4,5,2,6,3,1] => ? = 1

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: St001866
(load all 3 compositions to match this statistic)

Mapping

Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

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] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 2
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [1,3,4,2] => 0
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [4,3,1,2] => 1
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [3,4,2,1] => 1
[3,1,4,2] => [4,3,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 1
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [1,4,2,3] => 0
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [3,1,2,4] => 0
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[1,2,5,4,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 1
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[1,3,5,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 1
[1,3,5,4,2] => [1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 1
[1,4,2,5,3] => [1,5,4,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[1,4,3,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[1,4,5,3,2] => [1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[1,5,2,4,3] => [1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[1,5,3,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 1
[1,5,3,4,2] => [1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[1,5,4,2,3] => [1,4,2,5,3] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[1,5,4,3,2] => [1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
[2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => [3,2,5,1,4] => ? = 1
[2,1,5,4,3] => [2,1,4,5,3] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
[2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2
[2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 3
[2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[2,3,5,1,4] => [5,2,3,1,4] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[2,3,5,4,1] => [4,2,3,5,1] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[2,4,1,3,5] => [4,2,1,3,5] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 2
[2,4,1,5,3] => [5,2,4,1,3] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 2
[2,4,3,1,5] => [3,2,4,1,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3
[2,4,3,5,1] => [5,2,4,3,1] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[2,4,5,1,3] => [5,2,1,4,3] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[2,4,5,3,1] => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[2,5,1,3,4] => [5,2,1,3,4] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
[2,5,1,4,3] => [4,2,5,1,3] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1
[2,5,3,1,4] => [3,2,5,1,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 2
[2,5,3,4,1] => [4,2,5,3,1] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[2,5,4,1,3] => [4,2,1,5,3] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1
[2,5,4,3,1] => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
[3,1,4,2,5] => [4,3,1,2,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 2
[3,1,4,5,2] => [5,3,1,4,2] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 1
[3,1,5,2,4] => [5,3,1,2,4] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[3,1,5,4,2] => [4,3,1,5,2] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[3,2,1,4,5] => [2,3,1,4,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[3,2,1,5,4] => [2,3,1,5,4] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2
[3,2,4,5,1] => [5,3,2,4,1] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[3,2,5,4,1] => [4,3,2,5,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[3,4,2,5,1] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[3,4,5,2,1] => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[3,5,2,4,1] => [4,5,3,2,1] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[3,5,4,2,1] => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[4,2,3,5,1] => [5,4,2,3,1] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[4,2,5,3,1] => [3,5,4,2,1] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[4,3,2,5,1] => [5,3,4,2,1] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[4,3,5,2,1] => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[4,5,2,3,1] => [3,5,2,4,1] => [1,4,2,5,3] => [1,4,2,5,3] => 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 *

$i < j \leq \pi(j) < \pi(i)$ .

Matching statistic: St001882
(load all 3 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%

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] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[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] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 0
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 0
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 2
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[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] => [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[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
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [5,4,3,1,2] => ? = 0
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [5,4,1,3,2] => ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 0
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [5,2,4,1,3] => ? = 1
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 0
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 1
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[1,4,5,3,2] => [2,3,5,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 0
[1,5,3,2,4] => [4,2,3,5,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[1,5,3,4,2] => [2,4,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[1,5,4,2,3] => [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[1,5,4,3,2] => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 1
[2,1,4,3,5] => [5,3,4,1,2] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[2,1,4,5,3] => [3,5,4,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 1
[2,1,5,3,4] => [4,3,5,1,2] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 1
[2,1,5,4,3] => [3,4,5,1,2] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[2,3,1,4,5] => [5,4,1,3,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[2,3,1,5,4] => [4,5,1,3,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
[2,3,4,1,5] => [5,1,4,3,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,3,5,1,4] => [4,1,5,3,2] => [2,5,4,1,3] => [2,5,4,1,3] => ? = 2
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[2,4,1,3,5] => [5,3,1,4,2] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 2
[2,4,1,5,3] => [3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 2
[2,4,3,1,5] => [5,1,3,4,2] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,4,5,1,3] => [3,1,5,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[2,5,1,3,4] => [4,3,1,5,2] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
[2,5,1,4,3] => [3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 1
[2,5,3,1,4] => [4,1,3,5,2] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 2
[2,5,3,4,1] => [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[2,5,4,1,3] => [3,1,4,5,2] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
[2,5,4,3,1] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 1
[3,1,2,5,4] => [4,5,2,1,3] => [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
[3,1,4,2,5] => [5,2,4,1,3] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 2
[3,1,4,5,2] => [2,5,4,1,3] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 1
[3,1,5,2,4] => [4,2,5,1,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 1
[3,1,5,4,2] => [2,4,5,1,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 1
[3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[3,2,1,5,4] => [4,5,1,2,3] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,2,3,5,1] => [1,5,3,2,4] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[4,5,2,3,1] => [1,3,2,5,4] => [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.