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Your data matches 9 different statistics following compositions of up to 3 maps.
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Mp00066: Permutations inversePermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
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, adj0, counts adjacencies in the word with a zero appended. (adj0,des) and (fix,exc) are equidistributed, see [1].
Matching statistic: St000475
Mp00088: Permutations Kreweras complementPermutations
Mp00108: Permutations cycle typeInteger partitions
St000475: Integer partitions ⟶ ℤResult quality: 84% values known / values provided: 84%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
[8,5,3,4,2,6,7,1] => [1,6,4,5,3,7,8,2] => ?
=> ? = 1
[8,5,2,3,4,6,7,1] => [1,4,5,6,3,7,8,2] => ?
=> ? = 1
[5,6,4,3,7,2,8,1] => [1,7,5,4,2,3,6,8] => ?
=> ? = 3
[6,5,7,4,3,8,1,2] => [8,1,6,5,3,2,4,7] => ?
=> ? = 0
[7,3,4,5,6,8,1,2] => [8,1,3,4,5,6,2,7] => ?
=> ? = 4
[8,6,5,7,4,1,2,3] => [7,8,1,6,4,3,5,2] => ?
=> ? = 0
[8,7,3,4,2,5,1,6] => [8,6,4,5,7,1,3,2] => ?
=> ? = 0
[8,3,4,5,2,1,6,7] => [7,6,3,4,5,8,1,2] => ?
=> ? = 3
[6,7,2,3,4,1,5,8] => [7,4,5,6,8,2,3,1] => ?
=> ? = 0
[4,6,5,7,8,3,2,1] => [1,8,7,2,4,3,5,6] => ?
=> ? = 1
[2,5,8,7,6,4,3,1] => [1,2,8,7,3,6,5,4] => ?
=> ? = 3
[4,7,6,5,3,2,8,1] => [1,7,6,2,5,4,3,8] => ?
=> ? = 3
[2,5,4,7,6,3,8,1] => [1,2,7,4,3,6,5,8] => ?
=> ? = 5
[1,4,5,7,8,6,3,2] => [2,1,8,3,4,7,5,6] => ?
=> ? = 0
[1,4,5,7,6,8,3,2] => [2,1,8,3,4,6,5,7] => ?
=> ? = 1
[1,3,5,7,6,4,8,2] => [2,1,3,7,4,6,5,8] => ?
=> ? = 3
[1,3,6,5,7,4,8,2] => [2,1,3,7,5,4,6,8] => ?
=> ? = 3
[2,5,4,7,6,3,1,8] => [8,2,7,4,3,6,5,1] => ?
=> ? = 3
[2,5,4,6,3,7,1,8] => [8,2,6,4,3,5,7,1] => ?
=> ? = 3
[2,3,5,6,4,7,1,8] => [8,2,3,6,4,5,7,1] => ?
=> ? = 3
[2,3,4,6,5,7,1,8] => [8,2,3,4,6,5,7,1] => ?
=> ? = 4
[1,7,4,3,6,5,8,2] => [2,1,5,4,7,6,3,8] => ?
=> ? = 3
[1,7,4,5,6,3,8,2] => [2,1,7,4,5,6,3,8] => ?
=> ? = 4
[2,3,4,7,5,6,1,8] => [8,2,3,4,6,7,5,1] => ?
=> ? = 3
[2,7,4,3,6,5,8,1] => [1,2,5,4,7,6,3,8] => ?
=> ? = 5
[6,3,2,5,4,7,1,8] => [8,4,3,6,5,2,7,1] => ?
=> ? = 3
[2,4,1,5,6,8,3,7] => [4,2,8,3,5,6,1,7] => ?
=> ? = 3
[2,1,6,7,3,4,8,5] => [3,2,6,7,1,4,5,8] => ?
=> ? = 2
[2,3,4,6,1,7,8,5] => [6,2,3,4,1,5,7,8] => ?
=> ? = 5
[2,3,1,4,8,5,6,7] => [4,2,3,5,7,8,1,6] => ?
=> ? = 2
[3,1,4,6,7,2,8,5] => [3,7,2,4,1,5,6,8] => ?
=> ? = 2
[3,1,4,5,6,8,2,7] => [3,8,2,4,5,6,1,7] => ?
=> ? = 3
[1,3,2,6,8,4,5,7] => [2,4,3,7,8,5,1,6] => ?
=> ? = 1
[3,1,8,2,4,5,6,7] => [3,5,2,6,7,8,1,4] => ?
=> ? = 0
[1,3,4,5,8,2,6,7] => [2,7,3,4,5,8,1,6] => ?
=> ? = 3
[1,4,5,8,2,3,6,7] => [2,6,7,3,4,8,1,5] => ?
=> ? = 0
[1,4,5,2,3,6,7,8] => [2,5,6,3,4,7,8,1] => ?
=> ? = 0
[1,4,8,2,3,5,6,7] => [2,5,6,3,7,8,1,4] => ?
=> ? = 0
[1,5,6,7,2,3,4,8] => [2,6,7,8,3,4,5,1] => ?
=> ? = 0
[1,5,6,8,2,3,4,7] => [2,6,7,8,3,4,1,5] => ?
=> ? = 0
[1,2,3,6,7,4,5,8] => [2,3,4,7,8,5,6,1] => ?
=> ? = 0
[1,6,2,3,8,4,5,7] => [2,4,5,7,8,3,1,6] => ?
=> ? = 0
[1,2,3,6,8,4,5,7] => [2,3,4,7,8,5,1,6] => ?
=> ? = 0
[6,3,5,1,2,4,7,8] => [5,6,3,7,4,2,8,1] => ?
=> ? = 1
[8,7,3,6,1,2,4,5] => [6,7,4,8,1,5,3,2] => ?
=> ? = 0
[3,1,9,2,4,5,6,7,8] => [3,5,2,6,7,8,9,1,4] => ?
=> ? = 0
[7,1,6,8,4,3,5,2] => [3,1,7,6,8,4,2,5] => ?
=> ? = 0
[4,8,2,1,5,6,3,7] => [5,4,8,2,6,7,1,3] => ?
=> ? = 0
[4,7,1,8,6,2,3,5] => [4,7,8,2,1,6,3,5] => ?
=> ? = 1
[7,4,1,3,2,8,6,5] => [4,6,5,3,1,8,2,7] => ?
=> ? = 0
Description
The number of parts equal to 1 in a partition.
Mp00088: Permutations Kreweras complementPermutations
Mp00151: Permutations to cycle typeSet partitions
St000247: Set partitions ⟶ ℤResult quality: 59% values known / values provided: 59%distinct values known / distinct values provided: 73%
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
[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
[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
[8,7,5,6,3,4,2,1] => [1,8,6,7,4,5,3,2] => {{1},{2,8},{3,4,5,6,7}}
=> ? = 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
[8,3,4,5,6,7,2,1] => [1,8,3,4,5,6,7,2] => {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 6
[5,4,3,6,7,8,2,1] => [1,8,4,3,2,5,6,7] => {{1},{2,5,6,7,8},{3,4}}
=> ? = 1
[8,6,5,7,3,2,4,1] => [1,7,6,8,4,3,5,2] => {{1},{2,4,5,7,8},{3,6}}
=> ? = 1
[8,5,6,7,2,3,4,1] => [1,6,7,8,3,4,5,2] => {{1},{2,4,6,8},{3,5,7}}
=> ? = 1
[6,7,8,4,2,3,5,1] => [1,6,7,5,8,2,3,4] => {{1},{2,6},{3,7},{4,5,8}}
=> ? = 1
[8,5,3,4,2,6,7,1] => [1,6,4,5,3,7,8,2] => ?
=> ? = 1
[8,5,2,3,4,6,7,1] => [1,4,5,6,3,7,8,2] => ?
=> ? = 1
[8,4,3,2,5,6,7,1] => [1,5,4,3,6,7,8,2] => {{1},{2,5,6,7,8},{3,4}}
=> ? = 1
[8,3,4,2,5,6,7,1] => [1,5,3,4,6,7,8,2] => {{1},{2,5,6,7,8},{3},{4}}
=> ? = 3
[5,6,4,3,7,2,8,1] => [1,7,5,4,2,3,6,8] => ?
=> ? = 3
[4,5,6,7,2,3,8,1] => [1,6,7,2,3,4,5,8] => {{1},{2,4,6},{3,5,7},{8}}
=> ? = 2
[4,5,6,2,3,7,8,1] => [1,5,6,2,3,4,7,8] => {{1},{2,3,4,5,6},{7},{8}}
=> ? = 3
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[7,8,6,5,4,3,1,2] => [8,1,7,6,5,4,2,3] => {{1,2,3,7,8},{4,6},{5}}
=> ? = 1
[7,6,4,5,8,3,1,2] => [8,1,7,4,5,3,2,6] => {{1,2,3,6,7,8},{4},{5}}
=> ? = 2
[8,4,3,5,6,7,1,2] => [8,1,4,3,5,6,7,2] => {{1,2,8},{3,4},{5},{6},{7}}
=> ? = 3
[6,5,7,4,3,8,1,2] => [8,1,6,5,3,2,4,7] => ?
=> ? = 0
[7,5,4,3,6,8,1,2] => [8,1,5,4,3,6,2,7] => {{1,2,7,8},{3,5},{4},{6}}
=> ? = 2
[7,3,4,5,6,8,1,2] => [8,1,3,4,5,6,2,7] => ?
=> ? = 4
[8,7,6,5,4,2,1,3] => [8,7,1,6,5,4,3,2] => {{1,2,3,7,8},{4,6},{5}}
=> ? = 1
[7,6,8,5,4,2,1,3] => [8,7,1,6,5,3,2,4] => {{1,3,4,6,8},{2,7},{5}}
=> ? = 1
[6,7,8,5,4,1,2,3] => [7,8,1,6,5,2,3,4] => {{1,3,7},{2,4,6,8},{5}}
=> ? = 1
[8,6,5,7,4,1,2,3] => [7,8,1,6,4,3,5,2] => ?
=> ? = 0
[6,7,4,5,8,1,2,3] => [7,8,1,4,5,2,3,6] => {{1,3,7},{2,6,8},{4},{5}}
=> ? = 2
[4,5,6,7,8,1,2,3] => [7,8,1,2,3,4,5,6] => {{1,3,5,7},{2,4,6,8}}
=> ? = 0
[7,6,5,8,3,2,1,4] => [8,7,6,1,4,3,2,5] => {{1,4,5,8},{2,7},{3,6}}
=> ? = 0
[6,5,7,8,2,1,3,4] => [7,6,8,1,3,2,4,5] => {{1,4,7},{2,6},{3,5,8}}
=> ? = 0
[6,7,8,1,2,3,4,5] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> ? = 0
[8,7,3,4,2,5,1,6] => [8,6,4,5,7,1,3,2] => ?
=> ? = 0
[8,3,4,5,2,1,6,7] => [7,6,3,4,5,8,1,2] => {{1,7},{2,6,8},{3},{4},{5}}
=> ? = 3
[8,1,2,3,4,5,6,7] => [3,4,5,6,7,8,1,2] => {{1,3,5,7},{2,4,6,8}}
=> ? = 0
[5,6,7,4,3,2,1,8] => [8,7,6,5,2,3,4,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? = 0
[6,3,4,5,2,7,1,8] => [8,6,3,4,5,2,7,1] => {{1,8},{2,6},{3},{4},{5},{7}}
=> ? = 4
[4,3,2,5,6,7,1,8] => [8,4,3,2,5,6,7,1] => {{1,8},{2,4},{3},{5},{6},{7}}
=> ? = 4
[2,3,4,5,6,7,1,8] => [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 6
[6,7,2,3,4,1,5,8] => [7,4,5,6,8,2,3,1] => ?
=> ? = 0
[7,2,1,3,4,5,6,8] => [4,3,5,6,7,8,2,1] => {{1,4,6,8},{2,3,5,7}}
=> ? = 0
[6,2,3,4,5,1,7,8] => [7,3,4,5,6,2,8,1] => {{1,7,8},{2,3,4,5,6}}
=> ? = 0
[2,3,4,5,6,1,7,8] => [7,2,3,4,5,6,8,1] => {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 5
[6,1,2,3,4,5,7,8] => [3,4,5,6,7,2,8,1] => {{1,3,5,7,8},{2,4,6}}
=> ? = 0
[5,2,3,4,1,6,7,8] => [6,3,4,5,2,7,8,1] => {{1,6,7,8},{2,3,4,5}}
=> ? = 0
[2,3,4,5,1,6,7,8] => [6,2,3,4,5,7,8,1] => {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 4
[2,3,4,1,5,6,7,8] => [5,2,3,4,6,7,8,1] => {{1,5,6,7,8},{2},{3},{4}}
=> ? = 3
[4,6,5,7,8,3,2,1] => [1,8,7,2,4,3,5,6] => ?
=> ? = 1
Description
The number of singleton blocks of a set partition.
Mp00088: Permutations Kreweras complementPermutations
Mp00151: Permutations to cycle typeSet partitions
Mp00080: Set partitions to permutationPermutations
St000022: Permutations ⟶ ℤResult quality: 54% values known / values provided: 54%distinct values known / distinct values provided: 100%
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
[1,2,6,3,7,4,5] => [2,3,5,7,1,4,6] => {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 0
[1,3,2,4,5,6,7] => [2,4,3,5,6,7,1] => {{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => ? = 1
[1,3,4,5,7,6,2] => [2,1,3,4,5,7,6] => {{1,2},{3},{4},{5},{6,7}}
=> [2,1,3,4,5,7,6] => ? = 3
[1,3,4,6,5,7,2] => [2,1,3,4,6,5,7] => {{1,2},{3},{4},{5,6},{7}}
=> [2,1,3,4,6,5,7] => ? = 3
[1,3,4,7,5,6,2] => [2,1,3,4,6,7,5] => {{1,2},{3},{4},{5,6,7}}
=> [2,1,3,4,6,7,5] => ? = 2
[1,3,4,7,6,5,2] => [2,1,3,4,7,6,5] => {{1,2},{3},{4},{5,7},{6}}
=> [2,1,3,4,7,6,5] => ? = 3
[1,3,5,4,6,7,2] => [2,1,3,5,4,6,7] => {{1,2},{3},{4,5},{6},{7}}
=> [2,1,3,5,4,6,7] => ? = 3
[1,3,5,4,7,6,2] => [2,1,3,5,4,7,6] => {{1,2},{3},{4,5},{6,7}}
=> [2,1,3,5,4,7,6] => ? = 1
[1,3,5,6,4,7,2] => [2,1,3,6,4,5,7] => {{1,2},{3},{4,5,6},{7}}
=> [2,1,3,5,6,4,7] => ? = 2
[1,3,5,7,6,4,2] => [2,1,3,7,4,6,5] => {{1,2},{3},{4,5,7},{6}}
=> [2,1,3,5,7,6,4] => ? = 2
[1,3,6,5,4,7,2] => [2,1,3,6,5,4,7] => {{1,2},{3},{4,6},{5},{7}}
=> [2,1,3,6,5,4,7] => ? = 3
[1,3,6,5,7,4,2] => [2,1,3,7,5,4,6] => {{1,2},{3},{4,6,7},{5}}
=> [2,1,3,6,5,7,4] => ? = 2
[1,3,6,7,4,5,2] => [2,1,3,6,7,4,5] => {{1,2},{3},{4,6},{5,7}}
=> [2,1,3,6,7,4,5] => ? = 1
[1,3,7,5,6,4,2] => [2,1,3,7,5,6,4] => {{1,2},{3},{4,7},{5},{6}}
=> [2,1,3,7,5,6,4] => ? = 3
[1,3,7,6,5,4,2] => [2,1,3,7,6,5,4] => {{1,2},{3},{4,7},{5,6}}
=> [2,1,3,7,6,5,4] => ? = 1
[1,4,3,5,6,7,2] => [2,1,4,3,5,6,7] => {{1,2},{3,4},{5},{6},{7}}
=> [2,1,4,3,5,6,7] => ? = 3
[1,4,7,5,6,3,2] => [2,1,7,3,5,6,4] => {{1,2},{3,4,7},{5},{6}}
=> [2,1,4,7,5,6,3] => ? = 2
[1,4,7,6,3,5,2] => [2,1,6,3,7,5,4] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,5,2,3,4,7,6] => [2,4,5,6,3,1,7] => {{1,2,4,6},{3,5},{7}}
=> [2,4,5,6,3,1,7] => ? = 1
[1,5,4,7,6,3,2] => [2,1,7,4,3,6,5] => {{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => ? = 2
[1,5,7,3,2,4,6] => [2,6,5,7,3,1,4] => {{1,2,6},{3,5},{4,7}}
=> [2,6,5,7,3,1,4] => ? = 0
[1,6,4,5,7,3,2] => [2,1,7,4,5,3,6] => {{1,2},{3,6,7},{4},{5}}
=> [2,1,6,4,5,7,3] => ? = 2
[1,6,5,7,4,3,2] => [2,1,7,6,4,3,5] => {{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => ? = 0
[1,6,7,3,5,2,4] => [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => ? = 0
[1,7,2,3,4,5,6] => [2,4,5,6,7,1,3] => {{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => ? = 0
[1,7,2,3,5,6,4] => [2,4,5,1,6,7,3] => {{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => ? = 0
[1,7,2,4,3,6,5] => [2,4,6,5,1,7,3] => {{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => ? = 0
[1,7,4,5,6,3,2] => [2,1,7,4,5,6,3] => {{1,2},{3,7},{4},{5},{6}}
=> [2,1,7,4,5,6,3] => ? = 3
[2,1,3,7,4,5,6] => [3,2,4,6,7,1,5] => {{1,3,4,6},{2},{5,7}}
=> [3,2,4,6,7,1,5] => ? = 1
[2,1,6,3,4,5,7] => [3,2,5,6,7,4,1] => {{1,3,5,7},{2},{4,6}}
=> [3,2,5,6,7,4,1] => ? = 1
[2,3,4,5,1,7,6] => [6,2,3,4,5,1,7] => {{1,6},{2},{3},{4},{5},{7}}
=> [6,2,3,4,5,1,7] => ? = 5
[2,3,4,5,6,1,7] => [7,2,3,4,5,6,1] => {{1,7},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,1] => ? = 5
[2,3,4,6,5,1,7] => [7,2,3,4,6,5,1] => {{1,7},{2},{3},{4},{5,6}}
=> [7,2,3,4,6,5,1] => ? = 3
[2,3,5,4,6,1,7] => [7,2,3,5,4,6,1] => {{1,7},{2},{3},{4,5},{6}}
=> [7,2,3,5,4,6,1] => ? = 3
[2,3,6,4,5,1,7] => [7,2,3,5,6,4,1] => {{1,7},{2},{3},{4,5,6}}
=> [7,2,3,5,6,4,1] => ? = 2
[2,3,6,5,4,1,7] => [7,2,3,6,5,4,1] => {{1,7},{2},{3},{4,6},{5}}
=> [7,2,3,6,5,4,1] => ? = 3
[2,4,1,7,3,5,6] => [4,2,6,3,7,1,5] => {{1,3,4,6},{2},{5,7}}
=> [3,2,4,6,7,1,5] => ? = 1
[2,4,3,5,6,1,7] => [7,2,4,3,5,6,1] => {{1,7},{2},{3,4},{5},{6}}
=> [7,2,4,3,5,6,1] => ? = 3
[2,4,3,6,5,1,7] => [7,2,4,3,6,5,1] => {{1,7},{2},{3,4},{5,6}}
=> [7,2,4,3,6,5,1] => ? = 1
[2,4,5,3,6,1,7] => [7,2,5,3,4,6,1] => {{1,7},{2},{3,4,5},{6}}
=> [7,2,4,5,3,6,1] => ? = 2
[2,4,6,5,3,1,7] => [7,2,6,3,5,4,1] => {{1,7},{2},{3,4,6},{5}}
=> [7,2,4,6,5,3,1] => ? = 2
[2,5,4,3,6,1,7] => [7,2,5,4,3,6,1] => {{1,7},{2},{3,5},{4},{6}}
=> [7,2,5,4,3,6,1] => ? = 3
[2,5,4,6,3,1,7] => [7,2,6,4,3,5,1] => {{1,7},{2},{3,5,6},{4}}
=> [7,2,5,4,6,3,1] => ? = 2
[2,5,6,3,4,1,7] => [7,2,5,6,3,4,1] => {{1,7},{2},{3,5},{4,6}}
=> [7,2,5,6,3,4,1] => ? = 1
[2,6,4,5,3,1,7] => [7,2,6,4,5,3,1] => {{1,7},{2},{3,6},{4},{5}}
=> [7,2,6,4,5,3,1] => ? = 3
[2,6,5,4,3,1,7] => [7,2,6,5,4,3,1] => {{1,7},{2},{3,6},{4,5}}
=> [7,2,6,5,4,3,1] => ? = 1
[2,7,1,3,4,5,6] => [4,2,5,6,7,1,3] => {{1,4,6},{2},{3,5,7}}
=> [4,2,5,6,7,1,3] => ? = 1
[3,2,4,5,6,1,7] => [7,3,2,4,5,6,1] => {{1,7},{2,3},{4},{5},{6}}
=> [7,3,2,4,5,6,1] => ? = 3
[3,6,4,5,2,1,7] => [7,6,2,4,5,3,1] => {{1,7},{2,3,6},{4},{5}}
=> [7,3,6,4,5,2,1] => ? = 2
Description
The number of fixed points of a permutation.
Mp00089: Permutations Inverse Kreweras complementPermutations
Mp00151: Permutations to cycle typeSet partitions
Mp00221: Set partitions conjugateSet partitions
St000248: Set partitions ⟶ ℤResult quality: 48% values known / values provided: 48%distinct values known / distinct values provided: 82%
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] => [3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[2,1,3] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 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] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
[1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 2
[1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 0
[2,1,3,4] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 2
[2,3,1,4] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[2,4,1,3] => [1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[2,4,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[3,1,2,4] => [3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,1,4,2] => [4,1,3,2] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
[3,2,1,4] => [2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[3,2,4,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,4,2,1] => [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[4,1,3,2] => [4,3,1,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[4,2,1,3] => [2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[4,2,3,1] => [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[4,3,1,2] => [4,2,1,3] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 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] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 2
[1,2,5,3,4] => [2,4,5,3,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,4,3] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> 0
[1,3,2,4,5] => [3,2,4,5,1] => {{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> 1
[1,3,2,5,4] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> 2
[1,3,4,5,2] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> {{1},{2,3,4,5}}
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => {{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> {{1},{2,4,5},{3}}
=> 1
[1,4,2,3,5] => [3,4,2,5,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,4,2,5,3] => [3,5,2,4,1] => {{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,4,3,2,5] => [4,3,2,5,1] => {{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> 0
[1,4,3,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[1,4,5,2,3] => [4,5,2,3,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,4,5,3,2] => [5,4,2,3,1] => {{1,5},{2,3,4}}
=> {{1},{2,5},{3},{4}}
=> 0
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1,8] => {{1,7},{2,6},{3,5},{4},{8}}
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? = 2
[8,7,4,5,6,3,2,1] => [7,6,3,4,5,2,1,8] => {{1,7},{2,6},{3},{4},{5},{8}}
=> {{1,2},{3,8},{4,5,6,7}}
=> ? = 4
[8,4,5,6,7,3,2,1] => [7,6,2,3,4,5,1,8] => {{1,7},{2,3,4,5,6},{8}}
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 1
[8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1,8] => {{1,7},{2,3,4,5,6},{8}}
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 1
[5,6,7,8,3,4,2,1] => [7,5,6,1,2,3,4,8] => {{1,4,7},{2,5},{3,6},{8}}
=> {{1,2},{3,6},{4,7},{5,8}}
=> ? = 1
[5,4,3,6,7,8,2,1] => [7,3,2,1,4,5,6,8] => {{1,4,5,6,7},{2,3},{8}}
=> {{1,2},{3},{4},{5},{6,8},{7}}
=> ? = 1
[8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1,8] => {{1,2,3,5,6,7},{4},{8}}
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 2
[8,7,6,5,3,2,4,1] => [6,5,7,4,3,2,1,8] => {{1,2,3,5,6,7},{4},{8}}
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 2
[8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1,8] => {{1,3,4,6,7},{2,5},{8}}
=> {{1,2},{3},{4,7},{5,8},{6}}
=> ? = 1
[8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1,8] => {{1,3,5,7},{2,4,6},{8}}
=> {{1,2},{3,5,7},{4,6,8}}
=> ? = 1
[6,7,8,4,2,3,5,1] => [5,6,4,7,1,2,3,8] => {{1,5},{2,6},{3,4,7},{8}}
=> {{1,2,5},{3,7},{4,8},{6}}
=> ? = 1
[8,5,3,4,2,6,7,1] => [5,3,4,2,6,7,1,8] => ?
=> ?
=> ? = 1
[8,5,2,3,4,6,7,1] => [3,4,5,2,6,7,1,8] => {{1,3,5,6,7},{2,4},{8}}
=> {{1,2},{3},{4},{5,7},{6,8}}
=> ? = 1
[8,4,3,2,5,6,7,1] => [4,3,2,5,6,7,1,8] => {{1,4,5,6,7},{2,3},{8}}
=> {{1,2},{3},{4},{5},{6,8},{7}}
=> ? = 1
[8,3,4,2,5,6,7,1] => [4,2,3,5,6,7,1,8] => {{1,4,5,6,7},{2},{3},{8}}
=> {{1,2},{3},{4},{5},{6,7,8}}
=> ? = 3
[8,4,2,3,5,6,7,1] => [3,4,2,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[8,3,2,4,5,6,7,1] => [3,2,4,5,6,7,1,8] => {{1,3,4,5,6,7},{2},{8}}
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 2
[8,2,3,4,5,6,7,1] => [2,3,4,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[5,6,4,3,7,2,8,1] => [6,4,3,1,2,5,7,8] => {{1,2,4,5,6},{3},{7},{8}}
=> ?
=> ? = 3
[4,5,6,7,2,3,8,1] => [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 2
[4,5,6,2,3,7,8,1] => [4,5,1,2,3,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 3
[7,8,6,5,4,3,1,2] => [8,6,5,4,3,1,2,7] => {{1,2,6,7,8},{3,5},{4}}
=> {{1},{2},{3},{4,7},{5,6},{8}}
=> ? = 1
[7,6,4,5,8,3,1,2] => [8,6,3,4,2,1,5,7] => {{1,2,5,6,7,8},{3},{4}}
=> {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 2
[7,8,5,6,3,4,1,2] => [8,5,6,3,4,1,2,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,4,3,5,6,7,1,2] => [8,3,2,4,5,6,1,7] => {{1,7,8},{2,3},{4},{5},{6}}
=> ?
=> ? = 3
[6,5,7,4,3,8,1,2] => [8,5,4,2,1,3,6,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,5,4,3,6,8,1,2] => [8,4,3,2,5,1,6,7] => ?
=> ?
=> ? = 2
[7,3,4,5,6,8,1,2] => [8,2,3,4,5,1,6,7] => {{1,6,7,8},{2},{3},{4},{5}}
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 4
[3,4,5,6,7,8,1,2] => [8,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,6,5,4,2,1,3] => [6,8,5,4,3,2,1,7] => {{1,2,6,7,8},{3,5},{4}}
=> {{1},{2},{3},{4,7},{5,6},{8}}
=> ? = 1
[7,6,8,5,4,2,1,3] => [6,8,5,4,2,1,3,7] => {{1,6},{2,3,5,7,8},{4}}
=> {{1,4},{2},{3,8},{5,6},{7}}
=> ? = 1
[5,6,7,8,4,2,1,3] => [6,8,5,1,2,3,4,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,8,5,4,1,2,3] => [7,8,5,4,1,2,3,6] => {{1,3,5,7},{2,6,8},{4}}
=> {{1,3,7},{2,4,8},{5,6}}
=> ? = 1
[8,6,5,7,4,1,2,3] => [7,8,5,3,2,4,1,6] => ?
=> ?
=> ? = 0
[6,7,4,5,8,1,2,3] => [7,8,3,4,1,2,5,6] => {{1,5,7},{2,6,8},{3},{4}}
=> {{1,3,5,6,7},{2,4,8}}
=> ? = 2
[4,5,6,7,8,1,2,3] => [7,8,1,2,3,4,5,6] => {{1,3,5,7},{2,4,6,8}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 0
[7,6,5,8,3,2,1,4] => [6,5,8,3,2,1,4,7] => {{1,6},{2,5},{3,4,7,8}}
=> {{1,5},{2},{3,8},{4,7},{6}}
=> ? = 0
[7,5,6,8,2,3,1,4] => [5,6,8,2,3,1,4,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,5,8,3,1,2,4] => [7,5,8,3,1,2,4,6] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,5,7,8,2,1,3,4] => [5,7,8,2,1,3,4,6] => {{1,5},{2,4,7},{3,6,8}}
=> {{1,3,6},{2,5,7},{4,8}}
=> ? = 0
[5,6,7,8,1,2,3,4] => [6,7,8,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[7,8,6,1,2,3,4,5] => [5,6,7,8,3,1,2,4] => {{1,2,3,5,6,7},{4,8}}
=> {{1,5},{2,6},{3},{4},{7},{8}}
=> ? = 0
[6,7,8,1,2,3,4,5] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 0
[8,7,3,4,2,5,1,6] => [5,3,4,6,8,2,1,7] => {{1,5,7,8},{2,3,4,6}}
=> {{1},{2},{3,5},{4,8},{6},{7}}
=> ? = 0
[7,8,1,2,3,4,5,6] => [4,5,6,7,8,1,2,3] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,2,3,4,5,6,1,7] => [2,3,4,5,6,8,1,7] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,3,4,5,2,1,6,7] => [5,2,3,4,7,8,1,6] => ?
=> ?
=> ? = 3
[8,5,3,2,1,4,6,7] => [4,3,6,2,7,8,1,5] => ?
=> ?
=> ? = 0
[8,2,1,3,4,5,6,7] => [2,4,5,6,7,8,1,3] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 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.
St000241: Permutations ⟶ ℤResult quality: 40% values known / values provided: 40%distinct values known / distinct values provided: 64%
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
[1,2,3,6,7,4,5] => ? = 0
[1,2,6,3,7,4,5] => ? = 0
[1,3,2,4,5,6,7] => ? = 1
[1,3,4,5,6,7,2] => ? = 5
[1,3,4,5,7,6,2] => ? = 3
[1,3,4,6,5,7,2] => ? = 3
[1,3,4,7,5,6,2] => ? = 2
[1,3,4,7,6,5,2] => ? = 3
[1,3,5,4,6,7,2] => ? = 3
[1,3,5,4,7,6,2] => ? = 1
[1,3,5,6,4,7,2] => ? = 2
[1,3,5,7,6,4,2] => ? = 2
[1,3,6,5,4,7,2] => ? = 3
[1,3,6,5,7,4,2] => ? = 2
[1,3,6,7,4,5,2] => ? = 1
[1,3,7,5,6,4,2] => ? = 3
[1,3,7,6,5,4,2] => ? = 1
[1,4,3,5,6,7,2] => ? = 3
[1,4,7,5,6,3,2] => ? = 2
[1,4,7,6,3,5,2] => ? = 0
[1,5,2,3,4,7,6] => ? = 1
[1,5,4,7,6,3,2] => ? = 2
[1,5,7,3,2,4,6] => ? = 0
[1,6,2,3,7,4,5] => ? = 0
[1,6,2,7,4,3,5] => ? = 0
[1,6,4,5,7,3,2] => ? = 2
[1,6,5,7,4,3,2] => ? = 0
[1,6,7,3,4,2,5] => ? = 0
[1,6,7,3,5,2,4] => ? = 0
[1,7,2,3,4,5,6] => ? = 0
[1,7,2,3,4,6,5] => ? = 0
[1,7,2,3,5,4,6] => ? = 0
[1,7,2,3,5,6,4] => ? = 0
[1,7,2,4,3,5,6] => ? = 0
[1,7,2,4,3,6,5] => ? = 0
[1,7,3,2,4,5,6] => ? = 0
[1,7,4,5,6,3,2] => ? = 3
[2,1,3,4,5,6,7] => ? = 1
[2,1,3,4,7,5,6] => ? = 1
[2,1,3,7,4,5,6] => ? = 1
[2,1,4,5,6,7,3] => ? = 5
[2,1,6,3,4,5,7] => ? = 1
[2,1,7,3,4,5,6] => ? = 1
[2,3,1,4,5,6,7] => ? = 2
[2,3,4,1,5,6,7] => ? = 3
[2,3,4,5,1,6,7] => ? = 4
[2,3,4,5,1,7,6] => ? = 5
Description
The number of cyclical small excedances. A cyclical small excedance is an index i such that πi=i+1 considered cyclically.
Mp00089: Permutations Inverse Kreweras complementPermutations
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 27% values known / values provided: 27%distinct values known / distinct values provided: 64%
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] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,1,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 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] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 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] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[1,4,2,3] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[2,1,3,4] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,1,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[2,3,1,4] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 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] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[2,4,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,1,2,4] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[3,1,4,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[3,2,1,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[3,2,4,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[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] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 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] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[4,2,1,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[4,2,3,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[4,3,1,2] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[4,3,2,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 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] => [2,3,5,4,1] => [[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
[1,2,4,3,5] => [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
[1,2,4,5,3] => [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
[1,2,5,3,4] => [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
[1,2,5,4,3] => [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
[1,3,2,4,5] => [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,2,5,4] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[1,3,4,2,5] => [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 2
[1,3,4,5,2] => [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 3
[1,3,5,2,4] => [4,2,5,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 1
[1,3,5,4,2] => [5,2,4,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 1
[1,4,2,3,5] => [3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 0
[1,4,2,5,3] => [3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1
[1,4,3,2,5] => [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 0
[1,4,3,5,2] => [5,3,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
[1,4,5,2,3] => [4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 0
[1,2,4,6,3,5] => [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
[1,2,5,3,6,4] => [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
[1,2,5,6,3,4] => [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
[1,2,5,6,4,3] => [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
[1,3,4,6,2,5] => [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
[1,3,5,2,4,6] => [4,2,5,3,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,5,2,6,4] => [4,2,6,3,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,3,5,6,2,4] => [5,2,6,3,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,3,5,6,4,2] => [6,2,5,3,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1
[1,3,6,2,4,5] => [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,2,5,4] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,5,2,4] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,2,5,3,6] => [3,5,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,2,5,6,3] => [3,6,2,4,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[1,4,2,6,3,5] => [3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,4,2,6,5,3] => [3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,4,3,6,2,5] => [5,3,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,4,5,2,6,3] => [4,6,2,3,5,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,4,6,2,3,5] => [4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[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
[1,4,6,2,5,3] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,4,6,3,2,5] => [5,4,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,4,6,3,5,2] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 0
[1,5,2,3,6,4] => [3,4,6,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,2,4,6,3] => [3,6,4,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,5,2,6,3,4] => [3,5,6,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,5,2,6,4,3] => [3,6,5,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,5,3,2,6,4] => [4,3,6,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,3,6,2,4] => [5,3,6,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,5,3,6,4,2] => [6,3,5,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 0
[1,5,4,2,6,3] => [4,6,3,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[1,6,2,4,5,3] => [3,6,4,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,6,3,5,2,4] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,6,4,2,5,3] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
[2,1,4,6,3,5] => [1,5,3,6,4,2] => [[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],[0,0,0,1,0,0]]
=> ? = 2
[2,1,5,3,6,4] => [1,4,6,3,5,2] => [[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],[0,0,1,0,0,0]]
=> ? = 2
[2,1,5,6,3,4] => [1,5,6,3,4,2] => [[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],[0,0,1,0,0,0]]
=> ? = 1
[2,1,5,6,4,3] => [1,6,5,3,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1
[2,3,5,1,6,4] => [1,2,6,3,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 3
[2,3,6,1,5,4] => [1,2,6,5,3,4] => [[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],[0,0,1,0,0,0]]
=> ? = 2
[2,3,6,5,1,4] => [1,2,6,4,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 3
[2,4,1,3,6,5] => [1,4,2,6,5,3] => [[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],[0,0,0,1,0,0]]
=> ? = 2
[2,4,1,5,3,6] => [1,5,2,4,6,3] => [[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],[0,0,0,0,1,0]]
=> ? = 2
[2,4,1,5,6,3] => [1,6,2,4,5,3] => [[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],[0,1,0,0,0,0]]
=> ? = 3
[2,4,1,6,3,5] => [1,5,2,6,4,3] => [[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],[0,0,0,1,0,0]]
=> ? = 1
[2,4,1,6,5,3] => [1,6,2,5,4,3] => [[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],[0,1,0,0,0,0]]
=> ? = 1
[2,4,5,1,6,3] => [1,6,2,3,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[2,4,6,1,5,3] => [1,6,2,5,3,4] => [[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],[0,1,0,0,0,0]]
=> ? = 1
[2,4,6,5,1,3] => [1,6,2,4,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 2
[2,5,1,3,6,4] => [1,4,6,2,5,3] => [[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],[0,0,1,0,0,0]]
=> ? = 2
Description
The number of ones on the main diagonal of an alternating sign matrix.
Mp00088: Permutations Kreweras complementPermutations
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 64%
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.
Mp00089: Permutations Inverse Kreweras complementPermutations
Mp00305: Permutations parking functionParking functions
St001903: Parking functions ⟶ ℤResult quality: 2% values known / values provided: 2%distinct values known / distinct values provided: 45%
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 (a1,,an) is a parking function, a fixed point is an index i such that ai=i. It can be shown [1] that the generating function for parking functions with respect to this statistic is 1(n+1)2((q+n)n+1(q1)n+1).