- 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: St000356
(load all 4 compositions to match this statistic)

Mapping

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

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: St000223
(load all 3 compositions to match this statistic)

Mapping

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%

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

Description

The number of nestings in the permutation.

Matching statistic: St000039
(load all 2 compositions to match this statistic)

Mapping

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%

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

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: St000358
(load all 2 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
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
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [5,4,1,2,3] => 2
[1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [5,3,1,4,2] => 2
[1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,2,3,5] => [5,2,4,3,1] => 2
[1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 2
[1,4,3,5,2] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [5,2,1,4,3] => 2
[] => [] => ? = 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: St001866

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%

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

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%

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