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Matching statistic: St000463
(load all 2 compositions to match this statistic)

Mapping

St000463: 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] => 2
[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] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 0
[3,1,4,2] => 2
[3,2,1,4] => 0
[3,2,4,1] => 3
[3,4,1,2] => 4
[3,4,2,1] => 4
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 2
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 4
[1,4,5,2,3] => 4

Description

The number of admissible inversions of a permutation. Let

$w = w_1,w_2,\dots,w_k$ be a word of length

$k$ with distinct letters from

$[n]$ . An admissible inversion of

$w$ is a pair

$(w_i,w_j)$ such that

$1\leq i < j\leq k$ and

$w_i > w_j$ that satisfies either of the following conditions:

$1 < i$ and

$w_{i−1} < w_i$ or there is some

$l$ such that

$i < l < j$ and

$w_i < w_l$ .

Matching statistic: St000866

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000866: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 33%

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

Description

The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. An admissible inversion of a permutation

$\sigma$ is a pair

$(\sigma_i,\sigma_j)$ such that 1.

$i < j$ and

$\sigma_i > \sigma_j$ and 2. either

$\sigma_j < \sigma_{j+1}$ or there exists a

$i < k < j$ with

$\sigma_k < \sigma_j$ . This version was introduced by John Shareshian and Michelle L. Wachs in [1], for a closely related version, see [[St000463]].