- FindStat

Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St000039
(load all 7 compositions to match this statistic)

Mapping

St000039: 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] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 2
[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] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 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: St000223
(load all 6 compositions to match this statistic)

Mapping

Mp00239: Permutations —Corteel⟶ Permutations
St000223: 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] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => 2
[2,4,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [3,2,4,1] => 1
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [2,4,3,1] => 1
[3,4,1,2] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,1,2] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [3,1,4,2] => 0
[4,2,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => 1
[4,3,2,1] => [3,4,1,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,5,2,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 2

Description

The number of nestings in the permutation.

Matching statistic: St000371
(load all 6 compositions to match this statistic)

Mapping

Mp00239: Permutations —Corteel⟶ Permutations
St000371: 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] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => 2
[2,4,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [3,2,4,1] => 1
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [2,4,3,1] => 1
[3,4,1,2] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,1,2] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [3,1,4,2] => 0
[4,2,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => 1
[4,3,2,1] => [3,4,1,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,5,2,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 2

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: St001862
(load all 3 compositions to match this statistic)

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed 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] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 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] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [2,3,1,4] => 1
[2,3,4,1] => [2,3,4,1] => 2
[2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [4,1,3,2] => 0
[4,2,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,4,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,4,3,5,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 2

Description

The number of crossings of a signed permutation. A crossing of a signed permutation

$\pi$ is a pair

$(i, j)$ of indices such that *

$i < j \leq \pi(i) < \pi(j)$ , or *

$-i < j \leq -\pi(i) < \pi(j)$ , or *

$i > j > \pi(i) > \pi(j)$ .

Matching statistic: St000358
(load all 6 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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

Mapping

Mp00239: Permutations —Corteel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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 5 compositions to match this statistic)

Mapping

Mp00239: Permutations —Corteel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000356
(load all 2 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: 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] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => [1,2,3] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 2
[3,4,2,1] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[4,1,2,3] => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,2,1] => [2,1,3,4] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 0
[4,3,1,2] => [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [5,1,4,3,2] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 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: St001687
(load all 3 compositions to match this statistic)

Mapping

Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
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] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 0
[2,3,1] => [3,2,1] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [1,3,2] => 0
[3,2,1] => [2,3,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 0
[1,4,3,2] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 2
[2,4,1,3] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 1
[3,2,1,4] => [2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,4,3,1] => [3,4,2,1] => [2,1,3,4] => 1
[3,4,1,2] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 2
[3,4,2,1] => [4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 1
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 0
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[4,3,1,2] => [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,2,1] => [3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,3,5,4,2] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 0
[1,4,2,5,3] => [1,5,2,4,3] => [1,4,5,2,3] => [5,2,1,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [1,4,5,3,2] => [5,2,1,3,4] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 2

Description

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

Matching statistic: St001875

Mapping

Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%

Values

[1] => [1] => ([],1)
 => ([],1)
 => ? = 0 + 3
[1,2] => [1,2] => ([],2)
 => ([],1)
 => ? = 0 + 3
[2,1] => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => ? = 0 + 3
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 1 + 3
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => ? = 0 + 3
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 1 + 3
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 1 + 3
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 3
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[2,4,3,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,1,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 3
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 1 + 3
[3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 3
[3,4,2,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 3
[4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 0 + 3
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 1 + 3
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 1 + 3
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 2 + 3
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 3
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 1 + 3
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 3
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 3
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[1,5,2,4,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 0 + 3
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 3
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3

Description

The number of simple modules with projective dimension at most 1.

The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001857The number of edges in the reduced word graph of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000779The tier of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001964The interval resolution global dimension of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001645The pebbling number of a connected graph.