- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000475

Mapping

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
[1,2] => [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)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 62%

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
[8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
 => ? = 2
[7,8,6,5,4,3,2,1] => [1,8,7,6,5,4,2,3] => {{1},{2,3,7,8},{4,6},{5}}
 => ? = 2
[8,6,7,5,4,3,2,1] => [1,8,7,6,5,3,4,2] => {{1},{2,8},{3,4,6,7},{5}}
 => ? = 2
[6,7,8,5,4,3,2,1] => [1,8,7,6,5,2,3,4] => {{1},{2,4,6,8},{3,7},{5}}
 => ? = 2
[8,7,5,6,4,3,2,1] => [1,8,7,6,4,5,3,2] => {{1},{2,8},{3,7},{4,5,6}}
 => ? = 1
[7,8,5,6,4,3,2,1] => [1,8,7,6,4,5,2,3] => {{1},{2,3,7,8},{4,5,6}}
 => ? = 1
[8,6,5,7,4,3,2,1] => [1,8,7,6,4,3,5,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[7,5,6,8,4,3,2,1] => [1,8,7,6,3,4,2,5] => {{1},{2,3,5,7,8},{4,6}}
 => ? = 1
[5,6,7,8,4,3,2,1] => [1,8,7,6,2,3,4,5] => {{1},{2,5,8},{3,4,6,7}}
 => ? = 1
[8,7,6,4,5,3,2,1] => [1,8,7,5,6,4,3,2] => {{1},{2,8},{3,7},{4,5,6}}
 => ? = 1
[7,8,6,4,5,3,2,1] => [1,8,7,5,6,4,2,3] => {{1},{2,3,7,8},{4,5,6}}
 => ? = 1
[8,6,7,4,5,3,2,1] => [1,8,7,5,6,3,4,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[8,7,5,4,6,3,2,1] => [1,8,7,5,4,6,3,2] => {{1},{2,8},{3,7},{4,5},{6}}
 => ? = 2
[7,8,5,4,6,3,2,1] => [1,8,7,5,4,6,2,3] => {{1},{2,3,7,8},{4,5},{6}}
 => ? = 2
[8,7,4,5,6,3,2,1] => [1,8,7,4,5,6,3,2] => {{1},{2,8},{3,7},{4},{5},{6}}
 => ? = 4
[7,8,4,5,6,3,2,1] => [1,8,7,4,5,6,2,3] => {{1},{2,3,7,8},{4},{5},{6}}
 => ? = 4
[8,6,5,4,7,3,2,1] => [1,8,7,5,4,3,6,2] => {{1},{2,8},{3,6,7},{4,5}}
 => ? = 1
[8,5,6,4,7,3,2,1] => [1,8,7,5,3,4,6,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[8,6,4,5,7,3,2,1] => [1,8,7,4,5,3,6,2] => {{1},{2,8},{3,6,7},{4},{5}}
 => ? = 3
[8,5,4,6,7,3,2,1] => [1,8,7,4,3,5,6,2] => {{1},{2,8},{3,5,6,7},{4}}
 => ? = 2
[8,4,5,6,7,3,2,1] => [1,8,7,3,4,5,6,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[7,6,5,4,8,3,2,1] => [1,8,7,5,4,3,2,6] => {{1},{2,3,6,7,8},{4,5}}
 => ? = 1
[6,5,7,4,8,3,2,1] => [1,8,7,5,3,2,4,6] => {{1},{2,6,8},{3,4,5,7}}
 => ? = 1
[7,6,4,5,8,3,2,1] => [1,8,7,4,5,3,2,6] => {{1},{2,3,6,7,8},{4},{5}}
 => ? = 3
[6,7,4,5,8,3,2,1] => [1,8,7,4,5,2,3,6] => {{1},{2,6,8},{3,7},{4},{5}}
 => ? = 3
[6,5,4,7,8,3,2,1] => [1,8,7,4,3,2,5,6] => {{1},{2,6,8},{3,5,7},{4}}
 => ? = 2
[6,4,5,7,8,3,2,1] => [1,8,7,3,4,2,5,6] => {{1},{2,6,8},{3,4,5,7}}
 => ? = 1
[4,5,6,7,8,3,2,1] => [1,8,7,2,3,4,5,6] => {{1},{2,4,6,8},{3,5,7}}
 => ? = 1
[8,7,6,5,3,4,2,1] => [1,8,6,7,5,4,3,2] => {{1},{2,8},{3,4,6,7},{5}}
 => ? = 2
[8,6,7,5,3,4,2,1] => [1,8,6,7,5,3,4,2] => {{1},{2,8},{3,6},{4,7},{5}}
 => ? = 2
[7,6,8,5,3,4,2,1] => [1,8,6,7,5,3,2,4] => {{1},{2,4,7,8},{3,6},{5}}
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,8,6,7,4,5,3,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[8,5,6,7,3,4,2,1] => [1,8,6,7,3,4,5,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[6,5,7,8,3,4,2,1] => [1,8,6,7,3,2,4,5] => {{1},{2,3,5,6,8},{4,7}}
 => ? = 1
[5,6,7,8,3,4,2,1] => [1,8,6,7,2,3,4,5] => {{1},{2,5,8},{3,6},{4,7}}
 => ? = 1
[7,6,8,4,3,5,2,1] => [1,8,6,5,7,3,2,4] => {{1},{2,4,5,7,8},{3,6}}
 => ? = 1
[6,7,8,3,4,5,2,1] => [1,8,5,6,7,2,3,4] => {{1},{2,4,6,8},{3,5,7}}
 => ? = 1
[8,7,5,4,3,6,2,1] => [1,8,6,5,4,7,3,2] => {{1},{2,8},{3,6,7},{4,5}}
 => ? = 1
[8,7,4,5,3,6,2,1] => [1,8,6,4,5,7,3,2] => {{1},{2,8},{3,6,7},{4},{5}}
 => ? = 3
[8,7,5,3,4,6,2,1] => [1,8,5,6,4,7,3,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[8,7,4,3,5,6,2,1] => [1,8,5,4,6,7,3,2] => {{1},{2,8},{3,5,6,7},{4}}
 => ? = 2
[8,7,3,4,5,6,2,1] => [1,8,4,5,6,7,3,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
[8,6,5,4,3,7,2,1] => [1,8,6,5,4,3,7,2] => {{1},{2,8},{3,6},{4,5},{7}}
 => ? = 2
[8,6,4,5,3,7,2,1] => [1,8,6,4,5,3,7,2] => {{1},{2,8},{3,6},{4},{5},{7}}
 => ? = 4
[8,5,4,6,3,7,2,1] => [1,8,6,4,3,5,7,2] => {{1},{2,8},{3,5,6},{4},{7}}
 => ? = 3
[8,6,4,3,5,7,2,1] => [1,8,5,4,6,3,7,2] => {{1},{2,8},{3,5,6},{4},{7}}
 => ? = 3
[8,5,4,3,6,7,2,1] => [1,8,5,4,3,6,7,2] => {{1},{2,8},{3,5},{4},{6},{7}}
 => ? = 4
[8,5,3,4,6,7,2,1] => [1,8,4,5,3,6,7,2] => {{1},{2,8},{3,4,5},{6},{7}}
 => ? = 3
[8,4,3,5,6,7,2,1] => [1,8,4,3,5,6,7,2] => {{1},{2,8},{3,4},{5},{6},{7}}
 => ? = 4

Description

The number of singleton blocks of a set partition.

Matching statistic: St000022
(load all 22 compositions to match this statistic)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 85%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000248
(load all 2 compositions to match this statistic)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 69%

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
[8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
 => {{1,8},{2,7},{3,6},{4,5}}
 => ? = 2
[7,8,6,5,4,3,2,1] => [1,8,7,6,5,4,2,3] => {{1},{2,3,7,8},{4,6},{5}}
 => {{1,8},{2},{3,6},{4,5},{7}}
 => ? = 2
[8,6,7,5,4,3,2,1] => [1,8,7,6,5,3,4,2] => {{1},{2,8},{3,4,6,7},{5}}
 => {{1,8},{2,7},{3},{4,5},{6}}
 => ? = 2
[7,6,8,5,4,3,2,1] => [1,8,7,6,5,3,2,4] => {{1},{2,3,4,6,7,8},{5}}
 => {{1,8},{2},{3},{4,5},{6},{7}}
 => ? = 2
[6,7,8,5,4,3,2,1] => [1,8,7,6,5,2,3,4] => {{1},{2,4,6,8},{3,7},{5}}
 => {{1,8},{2,6},{3,7},{4,5}}
 => ? = 2
[8,7,5,6,4,3,2,1] => [1,8,7,6,4,5,3,2] => {{1},{2,8},{3,7},{4,5,6}}
 => {{1,8},{2,7},{3,6},{4},{5}}
 => ? = 1
[7,8,5,6,4,3,2,1] => [1,8,7,6,4,5,2,3] => {{1},{2,3,7,8},{4,5,6}}
 => ?
 => ? = 1
[8,6,5,7,4,3,2,1] => [1,8,7,6,4,3,5,2] => {{1},{2,8},{3,4,5,6,7}}
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 1
[7,6,5,8,4,3,2,1] => [1,8,7,6,4,3,2,5] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[7,5,6,8,4,3,2,1] => [1,8,7,6,3,4,2,5] => {{1},{2,3,5,7,8},{4,6}}
 => {{1,8},{2},{3,5},{4,6},{7}}
 => ? = 1
[6,5,7,8,4,3,2,1] => [1,8,7,6,3,2,4,5] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[5,6,7,8,4,3,2,1] => [1,8,7,6,2,3,4,5] => {{1},{2,5,8},{3,4,6,7}}
 => {{1,8},{2,5},{3},{4,7},{6}}
 => ? = 1
[8,7,6,4,5,3,2,1] => [1,8,7,5,6,4,3,2] => {{1},{2,8},{3,7},{4,5,6}}
 => {{1,8},{2,7},{3,6},{4},{5}}
 => ? = 1
[7,8,6,4,5,3,2,1] => [1,8,7,5,6,4,2,3] => {{1},{2,3,7,8},{4,5,6}}
 => ?
 => ? = 1
[8,6,7,4,5,3,2,1] => [1,8,7,5,6,3,4,2] => {{1},{2,8},{3,4,5,6,7}}
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 1
[8,7,5,4,6,3,2,1] => [1,8,7,5,4,6,3,2] => {{1},{2,8},{3,7},{4,5},{6}}
 => {{1,8},{2,7},{3,4,6},{5}}
 => ? = 2
[7,8,5,4,6,3,2,1] => [1,8,7,5,4,6,2,3] => {{1},{2,3,7,8},{4,5},{6}}
 => ?
 => ? = 2
[8,7,4,5,6,3,2,1] => [1,8,7,4,5,6,3,2] => {{1},{2,8},{3,7},{4},{5},{6}}
 => {{1,8},{2,7},{3,4,5,6}}
 => ? = 4
[7,8,4,5,6,3,2,1] => [1,8,7,4,5,6,2,3] => {{1},{2,3,7,8},{4},{5},{6}}
 => ?
 => ? = 4
[8,6,5,4,7,3,2,1] => [1,8,7,5,4,3,6,2] => {{1},{2,8},{3,6,7},{4,5}}
 => {{1,8},{2,7},{3},{4,6},{5}}
 => ? = 1
[8,5,6,4,7,3,2,1] => [1,8,7,5,3,4,6,2] => {{1},{2,8},{3,4,5,6,7}}
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 1
[8,6,4,5,7,3,2,1] => [1,8,7,4,5,3,6,2] => {{1},{2,8},{3,6,7},{4},{5}}
 => {{1,8},{2,7},{3},{4,5,6}}
 => ? = 3
[8,5,4,6,7,3,2,1] => [1,8,7,4,3,5,6,2] => {{1},{2,8},{3,5,6,7},{4}}
 => {{1,8},{2,7},{3},{4},{5,6}}
 => ? = 2
[8,4,5,6,7,3,2,1] => [1,8,7,3,4,5,6,2] => {{1},{2,8},{3,4,5,6,7}}
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 1
[7,6,5,4,8,3,2,1] => [1,8,7,5,4,3,2,6] => {{1},{2,3,6,7,8},{4,5}}
 => ?
 => ? = 1
[7,5,6,4,8,3,2,1] => [1,8,7,5,3,4,2,6] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[6,5,7,4,8,3,2,1] => [1,8,7,5,3,2,4,6] => {{1},{2,6,8},{3,4,5,7}}
 => ?
 => ? = 1
[5,6,7,4,8,3,2,1] => [1,8,7,5,2,3,4,6] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[7,6,4,5,8,3,2,1] => [1,8,7,4,5,3,2,6] => {{1},{2,3,6,7,8},{4},{5}}
 => {{1,8},{2},{3},{4,5,6},{7}}
 => ? = 3
[6,7,4,5,8,3,2,1] => [1,8,7,4,5,2,3,6] => {{1},{2,6,8},{3,7},{4},{5}}
 => ?
 => ? = 3
[7,5,4,6,8,3,2,1] => [1,8,7,4,3,5,2,6] => {{1},{2,3,5,6,7,8},{4}}
 => {{1,8},{2},{3},{4},{5,6},{7}}
 => ? = 2
[7,4,5,6,8,3,2,1] => [1,8,7,3,4,5,2,6] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[6,5,4,7,8,3,2,1] => [1,8,7,4,3,2,5,6] => {{1},{2,6,8},{3,5,7},{4}}
 => {{1,8},{2,4},{3,7},{5,6}}
 => ? = 2
[5,6,4,7,8,3,2,1] => [1,8,7,4,2,3,5,6] => {{1},{2,3,5,6,7,8},{4}}
 => {{1,8},{2},{3},{4},{5,6},{7}}
 => ? = 2
[6,4,5,7,8,3,2,1] => [1,8,7,3,4,2,5,6] => {{1},{2,6,8},{3,4,5,7}}
 => ?
 => ? = 1
[5,4,6,7,8,3,2,1] => [1,8,7,3,2,4,5,6] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[4,5,6,7,8,3,2,1] => [1,8,7,2,3,4,5,6] => {{1},{2,4,6,8},{3,5,7}}
 => {{1,8},{2,4,6},{3,5,7}}
 => ? = 1
[8,7,6,5,3,4,2,1] => [1,8,6,7,5,4,3,2] => {{1},{2,8},{3,4,6,7},{5}}
 => {{1,8},{2,7},{3},{4,5},{6}}
 => ? = 2
[7,8,6,5,3,4,2,1] => [1,8,6,7,5,4,2,3] => {{1},{2,3,4,6,7,8},{5}}
 => {{1,8},{2},{3},{4,5},{6},{7}}
 => ? = 2
[8,6,7,5,3,4,2,1] => [1,8,6,7,5,3,4,2] => {{1},{2,8},{3,6},{4,7},{5}}
 => ?
 => ? = 2
[7,6,8,5,3,4,2,1] => [1,8,6,7,5,3,2,4] => {{1},{2,4,7,8},{3,6},{5}}
 => {{1,8},{2},{3,6},{4,5,7}}
 => ? = 2
[6,7,8,5,3,4,2,1] => [1,8,6,7,5,2,3,4] => {{1},{2,3,4,6,7,8},{5}}
 => {{1,8},{2},{3},{4,5},{6},{7}}
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,8,6,7,4,5,3,2] => {{1},{2,8},{3,4,5,6,7}}
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 1
[7,8,5,6,3,4,2,1] => [1,8,6,7,4,5,2,3] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[8,5,6,7,3,4,2,1] => [1,8,6,7,3,4,5,2] => {{1},{2,8},{3,4,5,6,7}}
 => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 1
[6,7,5,8,3,4,2,1] => [1,8,6,7,4,2,3,5] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[7,5,6,8,3,4,2,1] => [1,8,6,7,3,4,2,5] => {{1},{2,3,4,5,6,7,8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 1
[6,5,7,8,3,4,2,1] => [1,8,6,7,3,2,4,5] => {{1},{2,3,5,6,8},{4,7}}
 => {{1,8},{2,5},{3,6},{4},{7}}
 => ? = 1
[5,6,7,8,3,4,2,1] => [1,8,6,7,2,3,4,5] => {{1},{2,5,8},{3,6},{4,7}}
 => {{1,8},{2,5},{3,6},{4,7}}
 => ? = 1

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

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 85%

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
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ? = 2
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 0
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 0
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => ? = 2
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,2,3,7,5,4,6] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => [1,2,3,6,7,5,4] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 0
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 0
[1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,2,3,7,5,6,4] => ? = 0
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 0
[1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [1,2,3,6,7,4,5] => ? = 0
[1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [1,2,3,5,7,4,6] => ? = 0
[1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => ? = 0
[1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [1,2,3,7,4,6,5] => ? = 1
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => ? = 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => ? = 3
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => ? = 3
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,5,4,3,7] => ? = 3
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => [1,2,7,5,4,3,6] => ? = 2
[1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => [1,2,6,7,5,4,3] => ? = 2
[1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,2,6,4,3,5,7] => ? = 1
[1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => ? = 2
[1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => [1,2,5,6,4,3,7] => ? = 1
[1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,2,5,7,6,4,3] => ? = 2
[1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,2,6,4,3,7,5] => ? = 1
[1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => [1,2,7,5,6,4,3] => ? = 1
[1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [1,2,7,4,3,5,6] => ? = 1
[1,2,4,7,3,6,5] => [1,2,5,3,7,6,4] => [1,2,6,7,4,3,5] => ? = 1
[1,2,4,7,5,3,6] => [1,2,6,3,5,7,4] => [1,2,5,7,4,3,6] => ? = 1
[1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => [1,2,5,6,7,4,3] => ? = 1
[1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [1,2,7,4,3,6,5] => ? = 2
[1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,2,6,5,7,4,3] => ? = 2
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 0
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => [1,2,6,5,3,4,7] => ? = 1
[1,2,5,3,6,7,4] => [1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => ? = 2
[1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [1,2,7,5,3,4,6] => ? = 0
[1,2,5,3,7,6,4] => [1,2,4,7,3,6,5] => [1,2,6,7,5,3,4] => ? = 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 7 compositions to match this statistic)

Mapping

St000241: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 54%

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

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: St000895
(load all 4 compositions to match this statistic)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 54%

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)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 54%

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)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001903: Parking functions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 38%

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

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