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: St000241
(load all 2 compositions to match this statistic)
Mapping
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: St000022
(load all 2 compositions to match this statistic)
Mapping
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [2,1,3] => 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] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => [3,4,2,1] => 0
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 0
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => [2,3,1,4] => 1
[2,4,1,3] => [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 0
[2,4,3,1] => [4,2,1,3] => [3,2,4,1] => [2,1,3,4] => 2
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => [3,1,4,2] => 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] => [1,2,4,3] => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 2
[3,4,1,2] => [2,4,3,1] => [4,1,3,2] => [4,3,1,2] => 0
[3,4,2,1] => [4,1,3,2] => [2,4,3,1] => [1,3,2,4] => 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] => [4,2,1,3] => 1
[4,2,1,3] => [3,1,4,2] => [2,4,1,3] => [1,4,2,3] => 1
[4,2,3,1] => [4,3,1,2] => [3,4,2,1] => [3,1,2,4] => 1
[4,3,1,2] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 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] => [2,3,5,4,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,1] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => [2,4,5,3,1] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => 0
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,1] => 0
[1,3,5,2,4] => [1,4,3,5,2] => [1,5,3,2,4] => [4,3,5,2,1] => 0
[1,3,5,4,2] => [1,5,3,2,4] => [1,4,3,5,2] => [5,3,2,4,1] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => [3,4,2,5,1] => 0
[1,4,2,5,3] => [1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => 0
[1,4,3,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 1
[1,4,5,2,3] => [1,3,5,4,2] => [1,5,2,4,3] => [3,5,4,2,1] => 0
Description
The number of fixed points of a permutation.
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: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 71%
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 $$ \frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right). $$