- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000356

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => 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: St000358

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000358: 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] => [1,2] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [3,2,1] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => [3,2,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [3,1,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => [3,2,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => [4,2,1,3] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => [4,1,3,2] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => [4,1,3,2] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [5,4,3,1,2] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => [5,4,2,3,1] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => [5,4,1,3,2] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => [5,4,1,3,2] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => [5,3,2,1,4] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [5,3,2,4,1] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,4,2] => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,4,2] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => [5,3,4,1,2] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => [5,2,1,4,3] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [5,2,4,3,1] => 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: St001727

Mapping

Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001727: 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
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => 0
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => [3,2,1] => 1
{{1},{2,3}}
 => {{1},{2,3}}
 => {{1},{2,3}}
 => [1,3,2] => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2,4},{3}}
 => {{1,2,4},{3}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{1,3,4},{2}}
 => {{1,3,4},{2}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 1
{{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 2
{{1},{2,3,4}}
 => {{1},{2,3,4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 1
{{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
 => {{1,2,3,5},{4}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => {{1,2,4},{3},{5}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 2
{{1,2},{3,4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2},{3,4,5}}
 => [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => {{1,2,5},{3},{4}}
 => {{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 2
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => {{1,3,4,5},{2}}
 => {{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => 1
{{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [3,5,4,1,2] => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 2
{{1,3},{2,5},{4}}
 => {{1,5},{2,3},{4}}
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 2
{{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
 => {{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => 2
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => 2

Description

The number of invisible inversions of a permutation. A visible inversion of a permutation

$\pi$ is a pair

$i < j$ such that

$\pi(j) \leq \min(i, \pi(i))$ . Thus, an invisible inversion satisfies

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

Matching statistic: St000497
(load all 2 compositions to match this statistic)

Mapping

St000497: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => ? = 0
{{1,2}}
 => 0
{{1},{2}}
 => 0
{{1,2,3}}
 => 0
{{1,2},{3}}
 => 0
{{1,3},{2}}
 => 1
{{1},{2,3}}
 => 0
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => 1
{{1,4},{2,3}}
 => 2
{{1},{2,3,4}}
 => 0
{{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => 0
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 0
{{1,2,3,4},{5}}
 => 0
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 0
{{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => 1
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 2
{{1,2},{3,4,5}}
 => 0
{{1,2},{3,4},{5}}
 => 0
{{1,2,5},{3},{4}}
 => 2
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => 1
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 1
{{1,3,5},{2},{4}}
 => 2
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => 2
{{1,4},{2,3,5}}
 => 2
{{1,4},{2,3},{5}}
 => 2

Description

The lcb statistic of a set partition. Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ . According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of

$S$ is given by a pair

$i < j$ such that

$j = \operatorname{max} B_b$ and

$i \in B_a$ for

$a > b$ .

Matching statistic: St000491
(load all 2 compositions to match this statistic)

Mapping

Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000491: Set partitions ⟶ 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
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => {{1},{2,3}}
 => 0
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 1
{{1},{2,3}}
 => {{1},{2,3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 0
{{1,2,4},{3}}
 => {{1,2,4},{3}}
 => {{1,3},{2,4}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 0
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 0
{{1,3,4},{2}}
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => 1
{{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => {{1,3,4},{2}}
 => 2
{{1},{2,3,4}}
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 2
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 1
{{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 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}}
 => {{1,3},{2,4,5}}
 => 1
{{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => 0
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 0
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => {{1,2,4},{3,5}}
 => 1
{{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => {{1,4},{2,3,5}}
 => 1
{{1,2,4},{3},{5}}
 => {{1,2,4},{3},{5}}
 => {{1},{2,4},{3,5}}
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => {{1,3,4},{2,5}}
 => 2
{{1,2},{3,4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => 0
{{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 0
{{1,2,5},{3},{4}}
 => {{1,2,5},{3},{4}}
 => {{1,4},{2},{3,5}}
 => 2
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => 0
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 0
{{1,3,4,5},{2}}
 => {{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => {{1,5},{2,3,4}}
 => 1
{{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => {{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => 1
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => {{1},{2,5},{3,4}}
 => 1
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => {{1,3},{2,5},{4}}
 => 2
{{1,3},{2,5},{4}}
 => {{1,5},{2,3},{4}}
 => {{1,5},{2},{3,4}}
 => 2
{{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => 1
{{1,3},{2},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => 1
{{1,4,5},{2,3}}
 => {{1,3},{2,4,5}}
 => {{1,2,4,5},{3}}
 => 2
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => {{1,4,5},{2,3}}
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => {{1},{2,4,5},{3}}
 => 2

Description

The number of inversions of a set partition. Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ . According to [1], see also [2,3], an inversion of

$S$ is given by a pair

$i > j$ such that

$j = \operatorname{min} B_b$ and

$i \in B_a$ for

$a < b$ . This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.

Matching statistic: St000609
(load all 2 compositions to match this statistic)

Mapping

Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ 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
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1,3},{2}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 1
{{1},{2,3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1,2,4},{3}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 1
{{1,2},{3,4}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 0
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1,3,4},{2}}
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => 1
{{1,3},{2,4}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 1
{{1,3},{2},{4}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 1
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => {{1},{2,3,4}}
 => 2
{{1},{2,3,4}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
{{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 2
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1
{{1},{2},{3,4}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 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},{4,5}}
 => 1
{{1,2,3},{4,5}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 0
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 0
{{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => 1
{{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => {{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => 2
{{1,2},{3,4,5}}
 => {{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => 0
{{1,2},{3,4},{5}}
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => 0
{{1,2,5},{3},{4}}
 => {{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 2
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => 0
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 0
{{1,3,4,5},{2}}
 => {{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => 1
{{1,3,4},{2,5}}
 => {{1,5},{2,3,4}}
 => {{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => {{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => 1
{{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => 2
{{1,3},{2,4,5}}
 => {{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => 1
{{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => 1
{{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => 2
{{1,3},{2,5},{4}}
 => {{1,5},{2,3},{4}}
 => {{1,3},{2},{4,5}}
 => 2
{{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => 1
{{1,3},{2},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => 1
{{1,4,5},{2,3}}
 => {{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => 2
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => 2
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => 2

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

Matching statistic: St000809

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000809: 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] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => [1,5,3,4,2] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2

Description

The reduced reflection length of the permutation. Let

$T$ be the set of reflections in a Coxeter group and let

$\ell(w)$ be the usual length function. Then the reduced reflection length of

$w$ is

$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$ In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].

Matching statistic: St001905

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 43%

Values

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

Description

The number of preferred parking spots in a parking function less than the index of the car. Let

$(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices

$1\leq i\leq n$ such that

$a_i < i$ .