Your data matches 14 different statistics following compositions of up to 3 maps.
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Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
St000021: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 1
[1,3,2] => [1,2,3] => [2,3,1] => 1
[2,1,3] => [1,2,3] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => 1
[3,1,2] => [1,3,2] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 2
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000157
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [2,1] => [[1],[2]]
=> 1
[2,1] => [1,2] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [2,3,1] => [[1,2],[3]]
=> 1
[1,3,2] => [1,2,3] => [2,3,1] => [[1,2],[3]]
=> 1
[2,1,3] => [1,2,3] => [2,3,1] => [[1,2],[3]]
=> 1
[2,3,1] => [1,2,3] => [2,3,1] => [[1,2],[3]]
=> 1
[3,1,2] => [1,3,2] => [2,1,3] => [[1,3],[2]]
=> 1
[3,2,1] => [1,3,2] => [2,1,3] => [[1,3],[2]]
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 2
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00069: Permutations complementPermutations
Mp00088: Permutations Kreweras complementPermutations
St000245: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [1,2] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,3,2] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 2
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 2
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 2
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 2
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 2
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000662
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
St000662: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[1,3,2] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,2,3,4,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,2,3,4,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [5,2,3,4,1] => 2
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000703
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000703: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [4,2,1,3] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [3,2,4,1] => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [3,2,4,1] => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [3,2,4,1] => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [3,2,4,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,2,3,1,4] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,2,3,1,4] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [5,2,3,1,4] => 2
Description
The number of deficiencies of a permutation. This is defined as $$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$ The number of exceedances is [[St000155]].
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
St000325: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => 2 = 1 + 1
[2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,3,2] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[2,1,3] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[3,2,1] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 2 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 2 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 2 + 1
[] => [] => [] => ? = 0 + 1
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
St000470: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => 2 = 1 + 1
[2,1] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,3,2] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[2,1,3] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[2,3,1] => [1,2,3] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[3,2,1] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 2 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 2 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 3 = 2 + 1
[] => [] => [] => ? = 0 + 1
Description
The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
St000155: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[1,3,2] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[2,1,3] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [3,2,1] => 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,2,3,1] => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [3,2,4,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [3,2,4,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [3,2,4,1] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [3,2,4,1] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [4,2,1,3] => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [4,2,1,3] => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [4,2,1,3] => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [4,2,1,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,2,3,4,1] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [3,2,1,4,5] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [4,2,3,5,1] => 2
[] => [] => [] => [] => ? = 0
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00131: Permutations descent bottomsBinary words
St000288: Binary words ⟶ ℤResult quality: 80% values known / values provided: 100%distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => => ? = 0
[1,2] => [1,2] => [2,1] => 1 => 1
[2,1] => [1,2] => [2,1] => 1 => 1
[1,2,3] => [1,2,3] => [2,3,1] => 10 => 1
[1,3,2] => [1,2,3] => [2,3,1] => 10 => 1
[2,1,3] => [1,2,3] => [2,3,1] => 10 => 1
[2,3,1] => [1,2,3] => [2,3,1] => 10 => 1
[3,1,2] => [1,3,2] => [2,1,3] => 10 => 1
[3,2,1] => [1,3,2] => [2,1,3] => 10 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => 100 => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => 100 => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 100 => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => 100 => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => 100 => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => 101 => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => 100 => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => 101 => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => 100 => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => 101 => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => 101 => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 101 => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => 100 => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => 101 => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => 100 => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => 100 => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => 100 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1000 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => 1000 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 1000 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => 1000 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => 1000 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1001 => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => 1000 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => 1001 => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => 1000 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => 1001 => 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => 1001 => 2
[] => [] => [] => => ? = 0
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000354
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00066: Permutations inversePermutations
St000354: Permutations ⟶ ℤResult quality: 80% values known / values provided: 100%distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[1,3,2] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [3,1,2] => 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 2
[] => [] => [] => [] => ? = 0
Description
The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001427The number of descents of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001946The number of descents in a parking function. St001890The maximum magnitude of the Möbius function of a poset.