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Your data matches 94 different statistics following compositions of up to 3 maps.
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Matching statistic: St000781
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St000781: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 1
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 1
[5,3]
=> 1
[5,2,1]
=> 2
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St000371
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 34%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 34%●distinct values known / distinct values provided: 29%
Values
[1]
=> [[1]]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => 0 = 1 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 0 = 1 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1 = 2 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0 = 1 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0 = 1 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => 0 = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => 0 = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => 0 = 1 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => 0 = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => 1 = 2 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0 = 1 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,3,4,2,6,7,5] => 0 = 1 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,2,7,3,4,5,6] => 0 = 1 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,4,7,2,6,3,5] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0 = 1 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => 0 = 1 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2] => ? = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3] => ? = 1 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 2 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 2 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,2,7,8,3,4,5,6] => ? = 3 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,4,7,8,2,6,3,5] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0 = 1 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 1 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,4,5,2,7,8,3,6] => ? = 3 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,3,8,2,5,4,7,6] => ? = 2 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,5,8,2,7,3,4,6] => ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0 = 1 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0 = 1 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => ? = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,2,5,6,7,8,9,3,4] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,4,5,6,7,8,9,2,3] => ? = 1 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 2 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 2 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,2,7,8,9,3,4,5,6] => ? = 3 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,4,7,8,9,2,6,3,5] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => ? = 1 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,3,4,5,2,7,8,9,6] => ? = 1 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,2,5,9,3,4,8,6,7] => ? = 2 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,4,5,9,2,7,8,3,6] => ? = 2 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,3,8,9,2,5,4,7,6] => ? = 2 - 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,5,8,9,2,7,3,4,6] => ? = 2 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,7,8,9,2,3,4,5,6] => ? = 1 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,3,6,2,5,9,4,8,7] => ? = 2 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,5,6,2,8,9,3,4,7] => ? = 3 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,2,9,3,4,5,6,7,8] => ? = 1 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,4,9,2,6,3,8,5,7] => ? = 2 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,6,9,2,8,3,4,5,7] => ? = 2 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 1 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,5,2,7,3,9,4,6,8] => ? = 1 - 1
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,7,2,9,3,4,5,6,8] => ? = 1 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,2] => ? = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,2,5,6,7,8,9,10,3,4] => ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [1,4,5,6,7,8,9,10,2,3] => ? = 1 - 1
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,3,7,8,9,10,4,5,6] => ? = 1 - 1
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9,10] => [1,3,6,7,8,9,10,2,5,4] => ? = 2 - 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [1,5,6,7,8,9,10,2,3,4] => ? = 1 - 1
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,4,9,10,5,6,7,8] => ? = 1 - 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [1,3,4,8,9,10,2,6,7,5] => ? = 2 - 1
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,2,7,8,9,10,3,4,5,6] => ? = 3 - 1
[6,2,1,1]
=> [[1,4,7,8,9,10],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9,10] => [1,4,7,8,9,10,2,6,3,5] => ? = 2 - 1
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [1,6,7,8,9,10,2,3,4,5] => ? = 1 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000119
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 29%
Values
[1]
=> [[1]]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => 0 = 1 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 0 = 1 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1 = 2 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0 = 1 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0 = 1 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => ? = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => ? = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => ? = 1 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => ? = 2 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0 = 1 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,3,4,2,6,7,5] => ? = 1 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,2,7,3,4,5,6] => ? = 1 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,4,7,2,6,3,5] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0 = 1 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => ? = 1 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2] => ? = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3] => ? = 1 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 2 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 2 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,2,7,8,3,4,5,6] => ? = 3 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,4,7,8,2,6,3,5] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0 = 1 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 1 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,4,5,2,7,8,3,6] => ? = 3 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,3,8,2,5,4,7,6] => ? = 2 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,5,8,2,7,3,4,6] => ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0 = 1 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => ? = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,2,5,6,7,8,9,3,4] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,4,5,6,7,8,9,2,3] => ? = 1 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 2 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 2 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,2,7,8,9,3,4,5,6] => ? = 3 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,4,7,8,9,2,6,3,5] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => ? = 1 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,3,4,5,2,7,8,9,6] => ? = 1 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,2,5,9,3,4,8,6,7] => ? = 2 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,4,5,9,2,7,8,3,6] => ? = 2 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,3,8,9,2,5,4,7,6] => ? = 2 - 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,5,8,9,2,7,3,4,6] => ? = 2 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,7,8,9,2,3,4,5,6] => ? = 1 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,3,6,2,5,9,4,8,7] => ? = 2 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,5,6,2,8,9,3,4,7] => ? = 3 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,2,9,3,4,5,6,7,8] => ? = 1 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,4,9,2,6,3,8,5,7] => ? = 2 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,6,9,2,8,3,4,5,7] => ? = 2 - 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [1,8,9,2,3,4,5,6,7] => 0 = 1 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 1 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,5,2,7,3,9,4,6,8] => ? = 1 - 1
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,7,2,9,3,4,5,6,8] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,2] => ? = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,2,5,6,7,8,9,10,3,4] => ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [1,4,5,6,7,8,9,10,2,3] => ? = 1 - 1
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000123
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 29%
Values
[1]
=> [[1]]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => 0 = 1 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 0 = 1 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1 = 2 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0 = 1 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0 = 1 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => ? = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => ? = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => ? = 1 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => ? = 2 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0 = 1 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,3,4,2,6,7,5] => ? = 1 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,2,7,3,4,5,6] => ? = 1 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,4,7,2,6,3,5] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0 = 1 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => ? = 1 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2] => ? = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3] => ? = 1 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 2 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 2 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,2,7,8,3,4,5,6] => ? = 3 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,4,7,8,2,6,3,5] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0 = 1 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 1 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,4,5,2,7,8,3,6] => ? = 3 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,3,8,2,5,4,7,6] => ? = 2 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,5,8,2,7,3,4,6] => ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0 = 1 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => ? = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,2,5,6,7,8,9,3,4] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,4,5,6,7,8,9,2,3] => ? = 1 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 2 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 2 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,2,7,8,9,3,4,5,6] => ? = 3 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,4,7,8,9,2,6,3,5] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => ? = 1 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,3,4,5,2,7,8,9,6] => ? = 1 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,2,5,9,3,4,8,6,7] => ? = 2 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,4,5,9,2,7,8,3,6] => ? = 2 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,3,8,9,2,5,4,7,6] => ? = 2 - 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,5,8,9,2,7,3,4,6] => ? = 2 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,7,8,9,2,3,4,5,6] => ? = 1 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,3,6,2,5,9,4,8,7] => ? = 2 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,5,6,2,8,9,3,4,7] => ? = 3 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,2,9,3,4,5,6,7,8] => ? = 1 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,4,9,2,6,3,8,5,7] => ? = 2 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,6,9,2,8,3,4,5,7] => ? = 2 - 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [1,8,9,2,3,4,5,6,7] => 0 = 1 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 1 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,5,2,7,3,9,4,6,8] => ? = 1 - 1
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,7,2,9,3,4,5,6,8] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,2] => ? = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,2,5,6,7,8,9,10,3,4] => ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [1,4,5,6,7,8,9,10,2,3] => ? = 1 - 1
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000223
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 29%
Values
[1]
=> [[1]]
=> [1] => [1] => 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0 = 1 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,4,2,3] => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 0 = 1 - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,4,5,2,3] => 0 = 1 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 0 = 1 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,5,2,3,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0 = 1 - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1 = 2 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0 = 1 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0 = 1 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => ? = 1 - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => ? = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => ? = 1 - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => ? = 2 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0 = 1 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,3,4,2,6,7,5] => ? = 1 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,2,7,3,4,5,6] => ? = 1 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,4,7,2,6,3,5] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0 = 1 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => ? = 1 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0 = 1 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0 = 1 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,3,4,5,6,7,8,2] => ? = 1 - 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,4,5,6,7,8,2,3] => ? = 1 - 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 2 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,5,6,7,8,2,3,4] => ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 2 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,2,7,8,3,4,5,6] => ? = 3 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,4,7,8,2,6,3,5] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0 = 1 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 1 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,4,5,2,7,8,3,6] => ? = 3 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,3,8,2,5,4,7,6] => ? = 2 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,5,8,2,7,3,4,6] => ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0 = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0 = 1 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,3,4,5,6,7,8,9,2] => ? = 1 - 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,2,5,6,7,8,9,3,4] => ? = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,4,5,6,7,8,9,2,3] => ? = 1 - 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 2 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,5,6,7,8,9,2,3,4] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 2 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,2,7,8,9,3,4,5,6] => ? = 3 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,4,7,8,9,2,6,3,5] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,6,7,8,9,2,3,4,5] => ? = 1 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,3,4,5,2,7,8,9,6] => ? = 1 - 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,2,5,9,3,4,8,6,7] => ? = 2 - 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,4,5,9,2,7,8,3,6] => ? = 2 - 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,3,8,9,2,5,4,7,6] => ? = 2 - 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,5,8,9,2,7,3,4,6] => ? = 2 - 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,7,8,9,2,3,4,5,6] => ? = 1 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,3,6,2,5,9,4,8,7] => ? = 2 - 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,5,6,2,8,9,3,4,7] => ? = 3 - 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,2,9,3,4,5,6,7,8] => ? = 1 - 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,4,9,2,6,3,8,5,7] => ? = 2 - 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,6,9,2,8,3,4,5,7] => ? = 2 - 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [1,8,9,2,3,4,5,6,7] => 0 = 1 - 1
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 1 - 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,5,2,7,3,9,4,6,8] => ? = 1 - 1
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,7,2,9,3,4,5,6,8] => ? = 1 - 1
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [1,3,4,5,6,7,8,9,10,2] => ? = 1 - 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,2,5,6,7,8,9,10,3,4] => ? = 1 - 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [1,4,5,6,7,8,9,10,2,3] => ? = 1 - 1
Description
The number of nestings in the permutation.
Matching statistic: St001084
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001084: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 29%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001084: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 29%
Values
[1]
=> [1,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [3,2,1] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,4,2,1,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,6,3,4] => 0 = 1 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,3,2,4] => 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,5,2,3,1,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,2,3,7,4,5] => 0 = 1 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,2,5,3] => 0 = 1 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [7,1,2,6,4,3,5] => ? = 1 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,5,1,2,6,4] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [7,1,6,3,4,2,5] => ? = 1 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [5,2,1,3,6,4] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [6,7,2,3,4,1,5] => ? = 1 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => [7,1,2,3,4,8,5,6] => 0 = 1 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [5,1,2,7,3,6,4] => ? = 1 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [8,1,2,3,7,5,4,6] => ? = 1 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,5,2,1,4,3] => 0 = 1 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [4,1,6,2,3,7,5] => ? = 2 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => [8,1,2,7,4,5,3,6] => ? = 1 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [5,1,6,2,3,4] => 0 = 1 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,6,1,4,5,2] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [3,6,1,2,4,7,5] => ? = 2 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [7,1,8,3,4,5,2,6] => ? = 1 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [5,2,3,1,6,4] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [7,2,1,3,4,6,5] => ? = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => [8,7,2,3,4,5,1,6] => ? = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,8,6] => 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 0 = 1 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => [8,1,2,3,4,5,9,6,7] => ? = 1 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => [6,1,2,3,8,4,7,5] => ? = 1 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,1,2,3,4,5,8,6,7] => [9,1,2,3,4,8,6,5,7] => ? = 1 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,1,6,3,2,5,4] => ? = 1 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,1,2,3,8,7,4,6] => [5,1,2,7,3,4,8,6] => ? = 2 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [9,1,2,3,4,8,5,6,7] => [9,1,2,3,8,5,6,4,7] => ? = 1 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,1,2,5,6,3] => 0 = 1 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,6,2,1,4,5,3] => ? = 2 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,2,5,6,3] => ? = 3 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,1,2,8,7,3,5,6] => [4,1,7,2,3,5,8,6] => ? = 2 - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,1,2,3,9,4,5,6,7] => [8,1,2,9,4,5,6,3,7] => ? = 1 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [4,6,5,3,1,2] => 0 = 1 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [7,1,6,2,3,5,4] => ? = 3 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,6,1,4,2,7,5] => ? = 2 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,7,8,2,4,5,6] => [3,8,1,2,4,5,7,6] => ? = 2 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,2,8,3,4,5,6,7] => [9,1,8,3,4,5,6,2,7] => ? = 1 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [5,1,6,3,4,2] => 0 = 1 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [6,2,3,1,4,7,5] => ? = 1 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [2,8,7,1,3,4,5,6] => [7,2,1,3,4,5,8,6] => ? = 1 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [9,1,8,2,3,4,5,6,7] => [9,8,2,3,4,5,6,1,7] => ? = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [9,1,2,3,4,5,6,8,7] => ? = 1 - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 0 = 1 - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => [9,1,2,3,4,5,6,10,7,8] => ? = 1 - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [7,1,2,3,4,5,8,9,6] => [7,1,2,3,4,9,5,8,6] => ? = 1 - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,1,2,3,4,5,6,9,7,8] => [10,1,2,3,4,5,9,7,6,8] => ? = 1 - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [8,1,2,3,6,7,4,5] => [8,1,2,7,4,3,6,5] => ? = 1 - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,1,2,3,4,9,8,5,7] => [6,1,2,3,8,4,5,9,7] => ? = 2 - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [10,1,2,3,4,5,9,6,7,8] => [10,1,2,3,4,9,6,7,5,8] => ? = 1 - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,5,2,3,6,1,4] => ? = 1 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [8,1,2,5,7,3,4,6] => [8,1,7,3,2,5,6,4] => ? = 2 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => [5,1,2,8,3,6,7,4] => ? = 3 - 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,1,2,3,9,8,4,6,7] => [5,1,2,8,3,4,6,9,7] => ? = 2 - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [9,1,2,3,4,10,5,6,7,8] => [9,1,2,3,10,5,6,7,4,8] => ? = 1 - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,1,2,7,3,6,5] => ? = 1 - 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,5,7,1,3,4,2] => ? = 2 - 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [7,1,4,8,2,3,5,6] => [7,8,2,1,4,5,6,3] => ? = 2 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => [4,1,7,2,5,3,8,6] => ? = 2 - 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,1,2,8,9,3,5,6,7] => [4,1,9,2,3,5,6,8,7] => ? = 2 - 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [10,1,2,3,9,4,5,6,7,8] => [10,1,2,9,4,5,6,7,3,8] => ? = 1 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => 0 = 1 - 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [4,6,7,3,1,5,2] => ? = 2 - 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [8,3,7,1,2,4,5,6] => [7,1,8,2,3,4,5,6] => ? = 3 - 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,6,2,1,3,4,5] => ? = 1 - 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => [3,7,1,4,2,5,8,6] => ? = 2 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,10,8] => 0 = 1 - 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [11,1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |1-23 in a permutation.
This is the number of occurrences of the pattern $123$, where the first two matched entries are the first two entries of the permutation.
In other words, this statistic is zero, if the first entry of the permutation is larger than the second, and it is the number of entries larger than the second entry otherwise.
Matching statistic: St000405
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 29%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000405: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 29%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0 = 1 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0 = 1 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 0 = 1 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => 0 = 1 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => 0 = 1 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? = 1 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => 0 = 1 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ? = 1 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 0 = 1 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => 1 = 2 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ? = 1 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0 = 1 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,3,2,4,5,1] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ? = 1 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => 0 = 1 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,8,7,1] => ? = 1 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => ? = 1 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,8,6,7,1] => ? = 1 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 0 = 1 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,7,5,4,6,1] => ? = 2 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => 0 = 1 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => 0 = 1 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 0 = 1 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,4,3,5,6,1] => ? = 2 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,8,4,5,6,7,1] => ? = 1 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [7,3,2,4,5,6,1] => ? = 1 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,8,3,4,5,6,7,1] => ? = 1 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8,2,3,4,5,6,7,1] => ? = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => 0 = 1 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,9,8,1] => ? = 1 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,8,7,6,1] => ? = 1 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,9,7,8,1] => ? = 1 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 1 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,4,8,6,5,7,1] => ? = 2 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,5,9,6,7,8,1] => ? = 1 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 0 = 1 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,7,4,5,3,6,1] => ? = 2 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 3 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,8,5,4,6,7,1] => ? = 2 - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,4,9,5,6,7,8,1] => ? = 1 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 0 = 1 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,4,2,5,6,1] => ? = 3 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,8,4,3,5,6,7,1] => ? = 2 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,3,9,4,5,6,7,8,1] => ? = 1 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 0 = 1 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [7,4,3,2,5,6,1] => ? = 1 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [8,3,2,4,5,6,7,1] => ? = 1 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,3,4,5,6,7,8,1] => ? = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,2,3,4,5,6,7,8,1] => ? = 1 - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => 0 = 1 - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,10,9,1] => ? = 1 - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,6,9,8,7,1] => ? = 1 - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,7,10,8,9,1] => ? = 1 - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,6,7,5,1] => ? = 1 - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,4,5,9,7,6,8,1] => ? = 2 - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,5,6,10,7,8,9,1] => ? = 1 - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 1 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,8,5,6,4,7,1] => ? = 2 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,8,7,6,5,1] => ? = 3 - 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,4,9,6,5,7,8,1] => ? = 2 - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,4,5,10,6,7,8,9,1] => ? = 1 - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [7,3,4,5,2,6,1] => ? = 1 - 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2 - 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [2,8,4,5,3,6,7,1] => ? = 2 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,8,6,5,4,7,1] => ? = 2 - 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,3,9,5,4,6,7,8,1] => ? = 2 - 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,3,4,10,5,6,7,8,9,1] => ? = 1 - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 0 = 1 - 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [7,3,5,4,2,6,1] => ? = 2 - 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 0 = 1 - 1
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000031
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 19%●distinct values known / distinct values provided: 14%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 19%●distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? = 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,2] => ? = 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,2] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? = 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,2] => ? = 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => ? = 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? = 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ? = 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,8,7,5,6,1,3,4] => ? = 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 3
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [2,8,4,6,1,7,3,5] => ? = 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ? = 3
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,1,8,7,3,6] => ? = 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => ? = 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ? = 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,9,2] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,9,4,5,6,7,8,1,3] => ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,10,2] => ? = 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,7,4,5,6,1,8,3] => ? = 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [2,9,8,5,6,7,1,3,4] => ? = 2
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,10,2] => ? = 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? = 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,8,6,5,1,7,3,4] => ? = 2
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => ? = 3
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [2,9,4,7,6,1,8,3,5] => ? = 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,10,2] => ? = 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 2
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [2,8,4,1,6,7,3,5] => ? = 2
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,4,7,6,1,3,5] => ? = 2
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [2,9,4,5,1,7,8,3,6] => ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000842
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000842: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 29%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 1 + 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 2 = 1 + 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 1 + 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 2 = 1 + 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [3,1,2] => 2 = 1 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 2 = 1 + 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,2,3,1] => 2 = 1 + 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [4,1,2,3] => 2 = 1 + 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,3,1,4] => 2 = 1 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,3,2,4] => 2 = 1 + 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,2,3,4,1] => 2 = 1 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,1,2,3,4] => 2 = 1 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,3,5,4,1] => 2 = 1 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => 2 = 1 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,5,4,2,1] => 2 = 1 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,3,2,5,4] => 2 = 1 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,4,3,2,5] => 2 = 1 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,2,3,4,5,1] => 2 = 1 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,1,2,3,4,5] => 2 = 1 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => 2 = 1 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,2,6,3,4,5] => 2 = 1 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,4,5,6,2,1] => 2 = 1 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [3,6,4,2,5,1] => 3 = 2 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,3,2,6,4,5] => 2 = 1 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,5,2,3,1,6] => 2 = 1 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [4,2,6,5,3,1] => 2 = 1 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,4,3,2,6,5] => 2 = 1 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,5,4,3,2,6] => 2 = 1 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,2,3,4,5,6,1] => ? = 1 + 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [7,1,2,3,4,5,6] => 2 = 1 + 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [2,3,7,4,5,6,1] => ? = 1 + 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,2,7,3,4,5,6] => 2 = 1 + 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [3,4,5,7,2,6,1] => ? = 1 + 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [3,7,4,2,5,6,1] => ? = 2 + 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,3,2,7,4,5,6] => 2 = 1 + 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [4,7,5,6,2,3,1] => ? = 1 + 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [4,5,2,3,7,6,1] => ? = 1 + 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [4,2,7,5,3,6,1] => ? = 2 + 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,4,3,2,7,5,6] => 2 = 1 + 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,3,6,7,4,2,1] => ? = 1 + 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [5,3,2,7,6,4,1] => ? = 1 + 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,5,4,3,2,7,6] => ? = 1 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,6,5,4,3,2,7] => ? = 1 + 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [8,2,3,4,5,6,7,1] => ? = 1 + 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [8,1,2,3,4,5,6,7] => 2 = 1 + 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [2,3,8,4,5,6,7,1] => ? = 1 + 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,2,8,3,4,5,6,7] => ? = 1 + 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [3,4,5,8,2,6,7,1] => ? = 1 + 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [3,8,4,2,5,6,7,1] => ? = 2 + 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,3,2,8,4,5,6,7] => 2 = 1 + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [4,5,6,7,8,2,3,1] => ? = 1 + 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [4,8,5,6,2,3,7,1] => ? = 2 + 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [4,5,2,3,8,6,7,1] => ? = 3 + 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [4,2,8,5,3,6,7,1] => ? = 2 + 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,4,3,2,8,5,6,7] => ? = 1 + 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [5,6,2,3,7,8,4,1] => ? = 1 + 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [5,2,8,6,7,3,4,1] => ? = 3 + 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [5,3,6,8,4,2,7,1] => ? = 2 + 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [5,3,2,8,6,4,7,1] => ? = 2 + 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,5,4,3,2,8,6,7] => ? = 1 + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [6,7,4,5,2,3,1,8] => ? = 1 + 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [6,4,2,7,8,5,3,1] => ? = 1 + 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [6,4,3,2,8,7,5,1] => ? = 1 + 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,6,5,4,3,2,8,7] => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,7,6,5,4,3,2,8] => ? = 1 + 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [9,2,3,4,5,6,7,8,1] => ? = 1 + 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [9,1,2,3,4,5,6,7,8] => 2 = 1 + 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [2,3,9,4,5,6,7,8,1] => ? = 1 + 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,2,9,3,4,5,6,7,8] => ? = 1 + 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [3,4,5,9,2,6,7,8,1] => ? = 1 + 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [3,9,4,2,5,6,7,8,1] => ? = 2 + 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,3,2,9,4,5,6,7,8] => ? = 1 + 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [4,5,6,7,9,2,3,8,1] => ? = 1 + 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [4,9,5,6,2,3,7,8,1] => ? = 2 + 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [4,5,2,3,9,6,7,8,1] => ? = 3 + 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [4,2,9,5,3,6,7,8,1] => ? = 2 + 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,4,3,2,9,5,6,7,8] => ? = 1 + 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [5,9,6,7,8,2,3,4,1] => ? = 1 + 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [5,6,2,3,7,9,4,8,1] => ? = 2 + 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [5,2,9,6,7,3,4,8,1] => ? = 2 + 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [5,3,6,9,4,2,7,8,1] => ? = 2 + 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [5,3,2,9,6,4,7,8,1] => ? = 2 + 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,5,4,3,2,9,6,7,8] => ? = 1 + 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [6,7,8,3,4,5,9,2,1] => ? = 1 + 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [10,1,2,3,4,5,6,7,8,9] => 2 = 1 + 1
Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
Matching statistic: St001344
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 29%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 29%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,5,2,1,6,4] => 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,1,5,3,6,7,4] => ? = 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,3,5,6,4,2,7] => 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,5,2,1,6,4,7] => ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,4,6,1,7,5,3] => ? = 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [4,6,3,2,1,7,5] => ? = 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? = 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,4,2,5,6,7,8] => ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => ? = 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,4,5,6,3,7,8] => ? = 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,4,1,5,3,6,7,8] => ? = 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? = 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => ? = 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,1,5,3,6,7,4,8] => ? = 2
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,3,5,6,4,2,7,8] => ? = 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [3,5,2,1,6,4,7,8] => ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => ? = 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,3,4,6,7,2,8,5] => ? = 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,2,6,1,4,7,8,5] => ? = 3
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [2,4,6,1,7,5,3,8] => ? = 2
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [4,6,3,2,1,7,5,8] => ? = 2
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => ? = 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => ? = 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [3,5,7,2,1,8,6,4] => ? = 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [5,7,4,3,2,1,8,6] => ? = 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => ? = 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? = 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? = 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => ? = 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,4,2,5,6,7,8,9] => ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => ? = 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,4,5,6,3,7,8,9] => ? = 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,4,1,5,3,6,7,8,9] => ? = 2
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => ? = 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,5,6,7,8,4,9] => ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,1,5,3,6,7,4,8,9] => ? = 2
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,3,5,6,4,2,7,8,9] => ? = 3
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [3,5,2,1,6,4,7,8,9] => ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => ? = 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,1,3,6,4,7,8,9,5] => ? = 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,3,4,6,7,2,8,5,9] => ? = 2
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [3,2,6,1,4,7,8,5,9] => ? = 2
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [2,4,6,1,7,5,3,8,9] => ? = 2
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000335The difference of lower and upper interactions. St000570The Edelman-Greene number of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000516The number of stretching pairs of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000095The number of triangles of a graph. St000664The number of right ropes of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000768The number of peaks in an integer composition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001260The permanent of an alternating sign matrix. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001429The number of negative entries in a signed permutation.
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