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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000241
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000241: 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] => 3
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 4
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 2
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[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] => 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 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2,1] => 0
[2,1] => [1,2] => 2
[1,2,3] => [2,3,1] => 0
[1,3,2] => [3,2,1] => 1
[2,1,3] => [1,3,2] => 1
[2,3,1] => [1,2,3] => 3
[3,1,2] => [3,1,2] => 0
[3,2,1] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [2,4,3,1] => 1
[1,3,2,4] => [3,2,4,1] => 1
[1,3,4,2] => [4,2,3,1] => 2
[1,4,2,3] => [3,4,2,1] => 0
[1,4,3,2] => [4,3,2,1] => 0
[2,1,3,4] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,3,2] => 2
[2,3,1,4] => [1,2,4,3] => 2
[2,3,4,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,4,2,3] => 1
[2,4,3,1] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => 0
[3,1,4,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,1,4,3] => 0
[3,2,4,1] => [2,1,3,4] => 2
[3,4,1,2] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => 1
[4,1,2,3] => [3,4,1,2] => 0
[4,1,3,2] => [4,3,1,2] => 0
[4,2,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,1,4] => 1
[4,3,1,2] => [4,2,1,3] => 1
[4,3,2,1] => [3,2,1,4] => 2
[1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,4,3,5] => [2,4,3,5,1] => 1
[1,2,4,5,3] => [2,5,3,4,1] => 2
[1,2,5,3,4] => [2,4,5,3,1] => 0
[1,2,5,4,3] => [2,5,4,3,1] => 0
[1,3,2,4,5] => [3,2,4,5,1] => 1
[1,3,2,5,4] => [3,2,5,4,1] => 2
[1,3,4,2,5] => [4,2,3,5,1] => 2
[1,3,4,5,2] => [5,2,3,4,1] => 3
[1,3,5,2,4] => [4,2,5,3,1] => 1
[1,3,5,4,2] => [5,2,4,3,1] => 1
[1,4,2,3,5] => [3,4,2,5,1] => 0
[1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,2,5] => [4,3,2,5,1] => 0
[1,4,3,5,2] => [5,3,2,4,1] => 1
[1,4,5,2,3] => [4,5,2,3,1] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000215
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: 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] => [3,1,2] => [3,2,1] => 3
[3,1,2] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,3,1] => 1
[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] => [1,4,2,3] => [1,4,3,2] => 2
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 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] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 4
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 1
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
[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] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 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] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 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] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 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: St000475
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%
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
[1,2] => [2,1] => [2]
=> 0
[2,1] => [1,2] => [1,1]
=> 2
[1,2,3] => [2,3,1] => [3]
=> 0
[1,3,2] => [2,1,3] => [2,1]
=> 1
[2,1,3] => [3,2,1] => [2,1]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 3
[3,1,2] => [3,1,2] => [3]
=> 0
[3,2,1] => [1,3,2] => [2,1]
=> 1
[1,2,3,4] => [2,3,4,1] => [4]
=> 0
[1,2,4,3] => [2,3,1,4] => [3,1]
=> 1
[1,3,2,4] => [2,4,3,1] => [3,1]
=> 1
[1,3,4,2] => [2,1,3,4] => [2,1,1]
=> 2
[1,4,2,3] => [2,4,1,3] => [4]
=> 0
[1,4,3,2] => [2,1,4,3] => [2,2]
=> 0
[2,1,3,4] => [3,2,4,1] => [3,1]
=> 1
[2,1,4,3] => [3,2,1,4] => [2,1,1]
=> 2
[2,3,1,4] => [4,2,3,1] => [2,1,1]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4
[2,4,1,3] => [4,2,1,3] => [3,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2
[3,1,2,4] => [3,4,2,1] => [4]
=> 0
[3,1,4,2] => [3,1,2,4] => [3,1]
=> 1
[3,2,1,4] => [4,3,2,1] => [2,2]
=> 0
[3,2,4,1] => [1,3,2,4] => [2,1,1]
=> 2
[3,4,1,2] => [4,1,2,3] => [4]
=> 0
[3,4,2,1] => [1,4,2,3] => [3,1]
=> 1
[4,1,2,3] => [3,4,1,2] => [2,2]
=> 0
[4,1,3,2] => [3,1,4,2] => [4]
=> 0
[4,2,1,3] => [4,3,1,2] => [4]
=> 0
[4,2,3,1] => [1,3,4,2] => [3,1]
=> 1
[4,3,1,2] => [4,1,3,2] => [3,1]
=> 1
[4,3,2,1] => [1,4,3,2] => [2,1,1]
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => [5]
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [4,1]
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => [4,1]
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => [3,1,1]
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => [5]
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => [3,2]
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => [4,1]
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => [3,1,1]
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => [3,1,1]
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => [4,1]
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,2,1]
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => [5]
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => [4,1]
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => [3,2]
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => [2,2,1]
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => [5]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000247
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> ? = 1
[1,2] => [2,1] => {{1,2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 2
[1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[1,3,2] => [2,1,3] => {{1,2},{3}}
=> 1
[2,1,3] => [3,2,1] => {{1,3},{2}}
=> 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 3
[3,1,2] => [3,1,2] => {{1,2,3}}
=> 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,4,2,3] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[3,1,2,4] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[3,1,4,2] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
[3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,4,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
[4,1,3,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,1,3] => [4,3,1,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,3,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => {{1,2,4,5},{3}}
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => {{1,2,3,4,5}}
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[1,4,5,3,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> {{1}}
=> ? = 1
[1,2] => [2,1] => {{1,2}}
=> {{1},{2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> {{1,2}}
=> 2
[1,2,3] => [2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,3,2] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[2,1,3] => [3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 3
[3,1,2] => [3,1,2] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,4,2,3] => [2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 2
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[3,1,2,4] => [3,4,2,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,1,4,2] => [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 0
[3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,4,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[4,1,3,2] => [3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[4,2,1,3] => [4,3,1,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[4,3,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
[4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,4,5,3,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000895
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1
[1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [1,2] => [[1,0],[0,1]]
=> 2
[1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[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] => [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] => [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] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[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] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[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] => [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] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[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
[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
[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
[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] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[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
[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
[3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[3,4,2,1] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[4,1,2,3] => [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] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[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
[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
[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
[4,3,2,1] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 0
[1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,2,6,4,3,5] => [2,3,6,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,2,6,5,3,4] => [2,3,6,1,5,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,3,2,6,4,5] => [2,4,3,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,3,4,6,2,5] => [2,6,3,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 2
[1,3,5,2,4,6] => [2,5,3,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,5,6,2,4] => [2,6,3,1,4,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1
[1,3,6,2,4,5] => [2,5,3,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,2,5,4] => [2,5,3,1,6,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,6,4,2,5] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
[1,3,6,5,2,4] => [2,6,3,1,5,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[1,4,2,5,3,6] => [2,4,6,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,4,2,6,3,5] => [2,4,6,3,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,4,3,6,2,5] => [2,6,4,3,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0
[1,4,5,2,3,6] => [2,5,6,3,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,4,5,3,2,6] => [2,6,5,3,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,4,6,2,3,5] => [2,5,6,3,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,4,6,3,2,5] => [2,6,5,3,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,4,6,5,2,3] => [2,6,1,3,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,4,6,5,3,2] => [2,1,6,3,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,3,6,2,4] => [2,6,4,1,3,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0
[1,5,4,6,2,3] => [2,6,1,4,3,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1
[1,5,4,6,3,2] => [2,1,6,4,3,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,5,6,3,2,4] => [2,6,5,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,5,6,4,2,3] => [2,6,1,5,3,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,5,6,4,3,2] => [2,1,6,5,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,6,2,4,3,5] => [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,6,2,5,3,4] => [2,4,6,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,6,2,5,4,3] => [2,4,1,6,5,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,6,3,2,4,5] => [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,6,3,2,5,4] => [2,5,4,1,6,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,6,3,4,2,5] => [2,6,4,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,6,3,5,2,4] => [2,6,4,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,6,4,2,3,5] => [2,5,6,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,6,4,2,5,3] => [2,5,1,4,6,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,6,4,3,2,5] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1
[1,6,4,5,2,3] => [2,6,1,4,5,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[1,6,5,2,3,4] => [2,5,6,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,6,5,2,4,3] => [2,5,1,6,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,6,5,3,2,4] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,6,5,4,2,3] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[2,1,3,6,4,5] => [3,2,4,6,1,5] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[2,1,4,6,3,5] => [3,2,6,4,1,5] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 2
[2,1,6,3,4,5] => [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[2,1,6,3,5,4] => [3,2,5,1,6,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[2,1,6,4,3,5] => [3,2,6,5,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[2,1,6,5,3,4] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[2,3,1,6,4,5] => [4,2,3,6,1,5] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 2
[2,3,5,1,4,6] => [5,2,3,6,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 2
[2,3,6,1,4,5] => [5,2,3,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 2
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000894
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> ? = 1
[1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [1,2] => [[1,0],[0,1]]
=> 2
[1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[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] => [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] => [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] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[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] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[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] => [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] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[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
[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
[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
[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] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[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
[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
[3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[3,4,2,1] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[4,1,2,3] => [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] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[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
[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
[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
[4,3,2,1] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 0
[1,4,5,3,2] => [2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 0
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,2,5,3,4,6] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,2,5,4,3,6] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,2,5,6,3,4] => [2,3,6,1,4,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,2,5,6,4,3] => [2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,2,6,3,4,5] => [2,3,5,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,2,6,3,5,4] => [2,3,5,1,6,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,6,4,3,5] => [2,3,6,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,6,5,3,4] => [2,3,6,1,5,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,2,6,5,4,3] => [2,3,1,6,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,2,4,5,6] => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,2,5,4,6] => [2,4,3,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,3,2,6,4,5] => [2,4,3,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,3,4,2,5,6] => [2,5,3,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 2
[1,3,4,5,2,6] => [2,6,3,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 3
[1,3,4,6,2,5] => [2,6,3,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 2
[1,3,5,2,4,6] => [2,5,3,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,5,4,2,6] => [2,6,3,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
[1,3,5,6,2,4] => [2,6,3,1,4,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1
[1,3,5,6,4,2] => [2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,2,4,5] => [2,5,3,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,2,5,4] => [2,5,3,1,6,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,6,4,2,5] => [2,6,3,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,6,5,2,4] => [2,6,3,1,5,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[1,3,6,5,4,2] => [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,4,2,3,5,6] => [2,4,5,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,4,2,5,3,6] => [2,4,6,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,4,2,6,3,5] => [2,4,6,3,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,4,3,2,5,6] => [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,4,3,5,2,6] => [2,6,4,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,4,3,6,2,5] => [2,6,4,3,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0
[1,4,5,2,3,6] => [2,5,6,3,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,4,5,3,2,6] => [2,6,5,3,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,4,5,6,2,3] => [2,6,1,3,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0
[1,4,5,6,3,2] => [2,1,6,3,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,4,6,2,3,5] => [2,5,6,3,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,4,6,2,5,3] => [2,5,1,3,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,4,6,3,2,5] => [2,6,5,3,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,4,6,3,5,2] => [2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,4,6,5,2,3] => [2,6,1,3,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,4,6,5,3,2] => [2,1,6,3,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,2,3,4,6] => [2,4,5,6,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
Description
The trace of an alternating sign matrix.
Matching statistic: St001903
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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%
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
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 2
[1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 2
[1,4,2,3] => [3,4,2,1] => [3,4,2,1] => 0
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 0
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 2
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 1
[4,3,1,2] => [4,2,1,3] => [4,2,1,3] => 1
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 2
[1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,2,5,3,4] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
[1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 1
[1,4,2,3,5] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1
[1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 0
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0
[1,5,2,3,4] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 0
[1,5,2,4,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 0
[1,5,3,2,4] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 0
[1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 0
[1,5,4,2,3] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => ? = 2
[2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => ? = 2
[2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 3
[2,1,5,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 1
[2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 2
[2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 3
[2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 3
[2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 2
[2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1
[2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 3
[2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
[2,5,1,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 1
[2,5,3,1,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
[2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 2
[2,5,4,1,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 2
[2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 3
[3,1,2,4,5] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 0
[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
$$
\frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right).
$$
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