searching the database
Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000215
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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
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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 0
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => 3
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => 0
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,3,4,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => 0
[2,3,4,1] => [4,2,3,1] => 1
[2,4,1,3] => [4,2,1,3] => 0
[2,4,3,1] => [3,2,4,1] => 2
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [3,4,1,2] => 0
[3,2,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [4,3,2,1] => 2
[3,4,1,2] => [4,1,3,2] => 0
[3,4,2,1] => [2,4,3,1] => 2
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [4,3,1,2] => 1
[4,2,1,3] => [2,4,1,3] => 1
[4,2,3,1] => [3,4,2,1] => 1
[4,3,1,2] => [3,1,4,2] => 1
[4,3,2,1] => [2,3,4,1] => 4
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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,2,1] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 2
[2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 0
[2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 1
[2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 0
[2,4,3,1] => [3,2,4,1] => [2,1,3,4] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 0
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => 0
[3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 2
[3,2,4,1] => [4,3,2,1] => [3,2,1,4] => 2
[3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 0
[3,4,2,1] => [2,4,3,1] => [1,3,2,4] => 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 0
[4,1,3,2] => [4,3,1,2] => [4,2,1,3] => 1
[4,2,1,3] => [2,4,1,3] => [1,4,2,3] => 1
[4,2,3,1] => [3,4,2,1] => [3,1,2,4] => 1
[4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 1
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 4
Description
The number of fixed points of a permutation.
Matching statistic: St000475
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ 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] => [2,1] => [2]
=> 0
[2,1] => [2,1] => [1,2] => [1,1]
=> 2
[1,2,3] => [1,2,3] => [2,3,1] => [3]
=> 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1]
=> 1
[2,1,3] => [2,1,3] => [3,2,1] => [2,1]
=> 1
[2,3,1] => [3,2,1] => [1,3,2] => [2,1]
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => [3]
=> 0
[3,2,1] => [2,3,1] => [1,2,3] => [1,1,1]
=> 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [3,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [2,2]
=> 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [4]
=> 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [3,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [2,1,1]
=> 2
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [2,2]
=> 0
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [3,1]
=> 1
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [4]
=> 0
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [2,1,1]
=> 2
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [4]
=> 0
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => [4]
=> 0
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [2,1,1]
=> 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [2,1,1]
=> 2
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [4]
=> 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [2,2]
=> 0
[4,1,3,2] => [4,3,1,2] => [4,1,3,2] => [3,1]
=> 1
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [3,1]
=> 1
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [3,1]
=> 1
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000895
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ 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] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [2,1] => [1,2] => [[1,0],[0,1]]
=> 2
[1,2,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,3] => [2,1,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,3,1] => [3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[4,1,3,2] => [4,3,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St001903
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001903: Parking functions ⟶ 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] => [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
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
$$
\frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right).
$$
Matching statistic: St000247
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> ? = 1
[1,2] => [1,2] => [2,1] => {{1,2}}
=> 0
[2,1] => [2,1] => [1,2] => {{1},{2}}
=> 2
[1,2,3] => [1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[1,3,2] => [1,3,2] => [2,1,3] => {{1,2},{3}}
=> 1
[2,1,3] => [2,1,3] => [3,2,1] => {{1,3},{2}}
=> 1
[2,3,1] => [3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => {{1,2,3}}
=> 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] => {{1,2,3,4}}
=> 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => {{1,2,3,4}}
=> 0
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
[4,1,3,2] => [4,3,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
Description
The number of singleton blocks of a set partition.
Matching statistic: St000894
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> ? = 1
[1,2] => [1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [2,1] => [1,2] => [[1,0],[0,1]]
=> 2
[1,2,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,3] => [2,1,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,3,1] => [3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[1,4,3,2] => [1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,4,1,3] => [4,2,1,3] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[2,4,3,1] => [3,2,4,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[3,2,1,4] => [2,3,1,4] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[3,2,4,1] => [4,3,2,1] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[3,4,1,2] => [4,1,3,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[4,1,3,2] => [4,3,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,2,1,3] => [2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[4,2,3,1] => [3,4,2,1] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[4,3,1,2] => [3,1,4,2] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
Description
The trace of an alternating sign matrix.
Matching statistic: St001060
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 2
[2,1] => ([],2)
=> ([],2)
=> ([],1)
=> ? = 2 + 2
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 2
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
[3,2,1] => ([],3)
=> ([],3)
=> ([],1)
=> ? = 3 + 2
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 2
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 2
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[4,3,2,1] => ([],4)
=> ([],4)
=> ([],1)
=> ? = 4 + 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001875
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 1 + 3
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 3
[2,1] => [2,1] => ([],2)
=> ([],1)
=> ? = 2 + 3
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 3
[3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 0 + 3
[3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? = 3 + 3
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 3
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 0 + 3
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 3
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? = 4 + 3
Description
The number of simple modules with projective dimension at most 1.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!