- FindStat

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000215
(load all 2 compositions to match this statistic)

Mapping

St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form

$(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic,

$\operatorname{adj_0}$ , counts adjacencies in the word with a zero appended.

$(\operatorname{adj_0}, \operatorname{des})$ and

$(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

Matching statistic: St000022
(load all 2 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 92%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000241
(load all 2 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 58%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index

$i$ such that

$\pi_i = i+1$ considered cyclically.

Matching statistic: St001903
(load all 4 compositions to match this statistic)

Mapping

Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001903: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 42%

Values

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

Description

The number of fixed points of a parking function. If

$(a_1,\dots,a_n)$ is a parking function, a fixed point is an index

$i$ such that

$a_i = i$ . It can be shown [1] that the generating function for parking functions with respect to this statistic is

$\frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right).$