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Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St001629
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => 0
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2,7],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2,7],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3,7],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,6],[2,4],[7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2,5],[7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3,5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,4,6,7],[2],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => [2,1] => 0
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2],[4],[7]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,4,6],[2,5,7],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,5,7],[4]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,6],[3,5,7],[4]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => 0
[[1,3,5],[2,6,7],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,5],[2,4,7],[6]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4],[2,5,7],[6]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4],[3,5,7],[6]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5],[2,4,6],[7]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,2] => 0
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,2] => 0
[[1,4,6],[2,5],[3,7]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,5],[4,7]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,6],[3,5],[4,7]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,4],[5,7]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000871
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Values
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,5,1,6,3] => 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,5,1,6,3] => 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,5,1,6,3] => 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [4,1,5,2,6,3] => 2 = 1 + 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [4,2,5,1,6,3] => 2 = 1 + 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [4,1,5,2,6,7,3] => ? = 0 + 1
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [4,1,5,6,2,7,3] => ? = 0 + 1
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,5,1,6,2,7,3] => ? = 1 + 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [4,2,5,1,6,7,3] => ? = 0 + 1
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [4,2,5,6,1,7,3] => ? = 0 + 1
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [4,5,2,6,1,7,3] => ? = 1 + 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [5,1,6,2,7,3,4] => ? = 0 + 1
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [4,1,5,2,6,7,3] => ? = 0 + 1
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [4,1,5,6,2,7,3] => ? = 0 + 1
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,5,1,6,2,7,3] => ? = 1 + 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [5,2,1,6,3,7,4] => ? = 1 + 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [5,2,6,7,1,3,4] => 1 = 0 + 1
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [5,6,2,7,1,3,4] => ? = 1 + 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [5,2,6,3,1,7,4] => ? = 0 + 1
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [5,2,6,1,7,3,4] => ? = 0 + 1
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [5,2,6,3,7,1,4] => ? = 0 + 1
[[1,3,5,6],[2,7],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [4,2,5,1,6,7,3] => ? = 0 + 1
[[1,3,4,6],[2,7],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [4,2,5,6,1,7,3] => ? = 0 + 1
[[1,2,4,6],[3,7],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [4,5,2,6,1,7,3] => ? = 1 + 1
[[1,3,5,6],[2,4],[7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [4,1,5,2,6,7,3] => ? = 0 + 1
[[1,3,4,6],[2,5],[7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [4,1,5,6,2,7,3] => ? = 0 + 1
[[1,2,4,6],[3,5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,5,1,6,2,7,3] => ? = 1 + 1
[[1,4,6,7],[2],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [5,3,2,6,1,7,4] => ? = 1 + 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [5,3,6,2,1,7,4] => ? = 0 + 1
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [5,3,6,2,7,1,4] => ? = 0 + 1
[[1,3,5,6],[2],[4],[7]]
=> [[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [4,2,5,1,6,7,3] => ? = 0 + 1
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [4,2,5,6,1,7,3] => ? = 0 + 1
[[1,2,4,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [4,5,2,6,1,7,3] => ? = 1 + 1
[[1,4,6],[2,5,7],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [5,2,1,6,3,7,4] => ? = 1 + 1
[[1,3,6],[2,5,7],[4]]
=> [[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [4,2,5,6,1,7,3] => ? = 0 + 1
[[1,2,6],[3,5,7],[4]]
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [4,5,2,6,1,7,3] => ? = 1 + 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [5,2,6,3,1,7,4] => ? = 0 + 1
[[1,3,5],[2,6,7],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [4,2,5,1,6,7,3] => ? = 0 + 1
[[1,3,5],[2,4,7],[6]]
=> [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [4,1,5,2,6,7,3] => ? = 0 + 1
[[1,3,4],[2,5,7],[6]]
=> [[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [4,1,5,6,2,7,3] => ? = 0 + 1
[[1,2,4],[3,5,7],[6]]
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,5,1,6,2,7,3] => ? = 1 + 1
[[1,3,5],[2,4,6],[7]]
=> [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [5,1,6,2,7,3,4] => ? = 0 + 1
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [5,2,6,3,7,1,4] => ? = 0 + 1
[[1,4,6],[2,5],[3,7]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => [5,2,1,6,3,7,4] => ? = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [5,2,6,7,1,3,4] => 1 = 0 + 1
[[1,2,6],[3,5],[4,7]]
=> [[1,2,3,5],[4,7],[6]]
=> [6,4,7,1,2,3,5] => [5,6,2,7,1,3,4] => ? = 1 + 1
[[1,3,6],[2,4],[5,7]]
=> [[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => [5,2,6,3,1,7,4] => ? = 0 + 1
[[1,3,5],[2,6],[4,7]]
=> [[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [5,2,6,1,7,3,4] => ? = 0 + 1
[[1,3,5],[2,4],[6,7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [4,1,5,2,6,7,3] => ? = 0 + 1
[[1,3,4],[2,5],[6,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [4,1,5,6,2,7,3] => ? = 0 + 1
[[1,2,4],[3,5],[6,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,5,1,6,2,7,3] => ? = 1 + 1
[[1,3,7],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [5,3,6,2,7,1,4] => ? = 0 + 1
[[1,4,6],[2,7],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [5,3,2,6,1,7,4] => ? = 1 + 1
[[1,3,6],[2,7],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [5,3,6,2,1,7,4] => ? = 0 + 1
[[1,3,5],[2,7],[4],[6]]
=> [[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [4,2,5,1,6,7,3] => ? = 0 + 1
[[1,3,4],[2,7],[5],[6]]
=> [[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [4,2,5,6,1,7,3] => ? = 0 + 1
[[1,2,4],[3,7],[5],[6]]
=> [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [4,5,2,6,1,7,3] => ? = 1 + 1
[[1,3,6],[2,5],[4],[7]]
=> [[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [5,2,6,7,1,3,4] => 1 = 0 + 1
[[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,5],[3,7],[6]]
=> [6,3,7,1,2,4,5] => [5,2,6,7,1,3,4] => 1 = 0 + 1
[[1,3,5,7,8],[2,4,6]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => [5,1,6,2,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7,8],[2,6],[4]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7,8],[2,4],[6]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,3,5,7,8],[2],[4],[6]]
=> [[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [5,3,6,2,7,1,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,4,6,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => [5,1,6,2,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,6,8],[4]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,4,8],[6]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,4,6],[8]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => [5,1,6,2,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,6],[4,8]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,4],[6,8]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,4,7,8],[2,6],[3],[5]]
=> [[1,2,5,6],[3,8],[4],[7]]
=> [7,4,3,8,1,2,5,6] => [6,3,2,7,8,1,4,5] => 2 = 1 + 1
[[1,3,5,7],[2,8],[4],[6]]
=> [[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [5,3,6,2,7,1,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,6],[4],[8]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5,7],[2,4],[6],[8]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,3,5,7],[2],[4],[6],[8]]
=> [[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [5,3,6,2,7,1,8,4] => 3 = 2 + 1
[[1,3,7],[2,4,8],[5,6]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,3,5],[2,6,8],[4,7]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5],[2,4,6],[7,8]]
=> [[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => [5,1,6,2,7,3,8,4] => 3 = 2 + 1
[[1,3,7],[2,6,8],[4],[5]]
=> [[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [5,3,6,2,7,1,8,4] => 3 = 2 + 1
[[1,3,5],[2,6,8],[4],[7]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5],[2,4,8],[6],[7]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,4,7],[2,6],[3,8],[5]]
=> [[1,2,5,6],[3,8],[4],[7]]
=> [7,4,3,8,1,2,5,6] => [6,3,2,7,8,1,4,5] => 2 = 1 + 1
[[1,3,5],[2,6],[4,8],[7]]
=> [[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [5,2,6,1,7,3,8,4] => 3 = 2 + 1
[[1,3,5],[2,4],[6,8],[7]]
=> [[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [5,2,6,3,7,1,8,4] => 3 = 2 + 1
[[1,3,5],[2,8],[4],[6],[7]]
=> [[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [5,3,6,2,7,1,8,4] => 3 = 2 + 1
[[1,4,7],[2,6],[3],[5],[8]]
=> [[1,2,5,6],[3,8],[4],[7]]
=> [7,4,3,8,1,2,5,6] => [6,3,2,7,8,1,4,5] => 2 = 1 + 1
[[1,4],[2,6],[3,8],[5],[7]]
=> [[1,2,5,6],[3,8],[4],[7]]
=> [7,4,3,8,1,2,5,6] => [6,3,2,7,8,1,4,5] => 2 = 1 + 1
[[1,5,8,9],[2,7],[3],[4],[6]]
=> [[1,2,6,7],[3,9],[4],[5],[8]]
=> [8,5,4,3,9,1,2,6,7] => [7,4,3,2,8,9,1,5,6] => 2 = 1 + 1
[[1,5],[2,7],[3,9],[4],[6],[8]]
=> [[1,2,6,7],[3,9],[4],[5],[8]]
=> [8,5,4,3,9,1,2,6,7] => [7,4,3,2,8,9,1,5,6] => 2 = 1 + 1
[[1,6,9,10],[2,8],[3],[4],[5],[7]]
=> [[1,2,7,8],[3,10],[4],[5],[6],[9]]
=> [9,6,5,4,3,10,1,2,7,8] => [8,5,4,3,2,9,10,1,6,7] => 2 = 1 + 1
[[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> [[1,2,7,8],[3,10],[4],[5],[6],[9]]
=> [9,6,5,4,3,10,1,2,7,8] => [8,5,4,3,2,9,10,1,6,7] => 2 = 1 + 1
[[1,5,8],[2,7],[3],[4],[6],[9]]
=> [[1,2,6,7],[3,9],[4],[5],[8]]
=> [8,5,4,3,9,1,2,6,7] => [7,4,3,2,8,9,1,5,6] => 2 = 1 + 1
[[1,6,9],[2,8],[3],[4],[5],[7],[10]]
=> [[1,2,7,8],[3,10],[4],[5],[6],[9]]
=> [9,6,5,4,3,10,1,2,7,8] => [8,5,4,3,2,9,10,1,6,7] => 2 = 1 + 1
[[1,5,8],[2,7],[3,9],[4],[6]]
=> [[1,2,6,7],[3,9],[4],[5],[8]]
=> [8,5,4,3,9,1,2,6,7] => [7,4,3,2,8,9,1,5,6] => 2 = 1 + 1
[[1,6,9],[2,8],[3,10],[4],[5],[7]]
=> [[1,2,7,8],[3,10],[4],[5],[6],[9]]
=> [9,6,5,4,3,10,1,2,7,8] => [8,5,4,3,2,9,10,1,6,7] => 2 = 1 + 1
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St000447
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000447: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000447: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6,7],[2,4]]
=> [1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6,7],[2,5]]
=> [1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6,7],[3,5]]
=> [2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5,6,7],[2],[4]]
=> [1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6,7],[2],[5]]
=> [1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6,7],[3],[5]]
=> [2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5,7],[2,4,6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,6],[2,4,7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2,5,7]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3,5,7]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6,7],[3,5],[4]]
=> [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6,7],[2,4],[5]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,7],[2,6],[4]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,7],[2,4],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,6],[2,7],[4]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2,7],[5]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3,7],[5]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5,6],[2,4],[7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2,5],[7]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3,5],[7]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,7],[2],[4],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,6],[2],[4],[7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2],[5],[7]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3],[5],[7]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6],[3,5,7],[4]]
=> [2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6],[2,4,7],[5]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,6,7],[4]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,4,7],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4],[2,5,7],[6]]
=> [1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4],[3,5,7],[6]]
=> [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5],[2,4,6],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,7],[2,4],[5,6]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4,6],[2,5],[3,7]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,5],[4,7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6],[3,5],[4,7]]
=> [2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6],[2,4],[5,7]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,6],[4,7]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,4],[6,7]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4],[2,5],[6,7]]
=> [1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4],[3,5],[6,7]]
=> [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,7],[2,6],[4],[5]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4,6],[2,7],[3],[5]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,7],[4],[5]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,7],[4],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4],[2,7],[5],[6]]
=> [1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4],[3,7],[5],[6]]
=> [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,5],[4],[7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6],[3,5],[4],[7]]
=> [2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6],[2,4],[5],[7]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,6],[4],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,4],[6],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3],[2,6],[4,7],[5]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4],[2,5],[3,7],[6]]
=> [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,2],[3,5],[4,7],[6]]
=> [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,7,8],[3,6],[4],[5]]
=> [2,1,1,4] => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,7],[3,6,8],[4],[5]]
=> [2,1,1,3,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,5],[3,6,8],[4],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,4],[3,5,8],[6],[7]]
=> [2,2,1,1,2] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,7],[3,6],[4,8],[5]]
=> [2,1,1,3,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,5],[3,6],[4,8],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,4],[3,5],[6,8],[7]]
=> [2,2,1,1,2] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,5],[3,8],[4],[6],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,4],[3,8],[5],[6],[7]]
=> [2,2,1,1,2] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,7],[3,6],[4],[5],[8]]
=> [2,1,1,3,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2],[3,5],[4,6],[7,8]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2],[3,6],[4,8],[5],[7]]
=> [2,1,1,2,2] => [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2],[3,5],[4,8],[6],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
Description
The number of pairs of vertices of a graph with distance 3.
This is the coefficient of the cubic term of the Wiener polynomial, also called Wiener polarity index.
Matching statistic: St000449
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000449: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000449: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[[1,3,5,6,7],[2,4]]
=> [1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6,7],[2,5]]
=> [1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6,7],[3,5]]
=> [2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5,6,7],[2],[4]]
=> [1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6,7],[2],[5]]
=> [1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6,7],[3],[5]]
=> [2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5,7],[2,4,6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,6],[2,4,7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2,5,7]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3,5,7]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6,7],[2,5],[4]]
=> [1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6,7],[3,5],[4]]
=> [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6,7],[2,4],[5]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,7],[2,6],[4]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,7],[2,4],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,6],[2,7],[4]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2,7],[5]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3,7],[5]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5,6],[2,4],[7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2,5],[7]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3,5],[7]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,7],[2],[4],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5,6],[2],[4],[7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4,6],[2],[5],[7]]
=> [1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4,6],[3],[5],[7]]
=> [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6],[3,5,7],[4]]
=> [2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6],[2,4,7],[5]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,6,7],[4]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,4,7],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4],[2,5,7],[6]]
=> [1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4],[3,5,7],[6]]
=> [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,5],[2,4,6],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,7],[2,4],[5,6]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4,6],[2,5],[3,7]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,5],[4,7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6],[3,5],[4,7]]
=> [2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6],[2,4],[5,7]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,6],[4,7]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,4],[6,7]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4],[2,5],[6,7]]
=> [1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4],[3,5],[6,7]]
=> [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,7],[2,6],[4],[5]]
=> [1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4,6],[2,7],[3],[5]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,7],[4],[5]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,7],[4],[6]]
=> [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,4],[2,7],[5],[6]]
=> [1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,4],[3,7],[5],[6]]
=> [2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2,5],[4],[7]]
=> [1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,2,6],[3,5],[4],[7]]
=> [2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,3,6],[2,4],[5],[7]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,6],[4],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2,4],[6],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,3],[2,6],[4,7],[5]]
=> [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 - 1
[[1,4],[2,5],[3,7],[6]]
=> [1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[[1,2],[3,5],[4,7],[6]]
=> [2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[[1,2,7,8],[3,6],[4],[5]]
=> [2,1,1,4] => [1,1,1,4,1] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,7],[3,6,8],[4],[5]]
=> [2,1,1,3,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,5],[3,6,8],[4],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,4],[3,5,8],[6],[7]]
=> [2,2,1,1,2] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,7],[3,6],[4,8],[5]]
=> [2,1,1,3,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,5],[3,6],[4,8],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,4],[3,5],[6,8],[7]]
=> [2,2,1,1,2] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,5],[3,8],[4],[6],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,4],[3,8],[5],[6],[7]]
=> [2,2,1,1,2] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2,7],[3,6],[4],[5],[8]]
=> [2,1,1,3,1] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2],[3,5],[4,6],[7,8]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2],[3,6],[4,8],[5],[7]]
=> [2,1,1,2,2] => [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
[[1,2],[3,5],[4,8],[6],[7]]
=> [2,1,2,1,2] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 0 = 1 - 1
Description
The number of pairs of vertices of a graph with distance 4.
This is the coefficient of the quartic term of the Wiener polynomial.
Matching statistic: St000742
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Mp00066: Permutations —inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000742: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 40%
Values
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [2,1,4,3,5,6] => 3 = 1 + 2
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [3,2,4,1,5,6] => [2,1,4,3,5,6] => 3 = 1 + 2
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,5,2,6,3] => [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,5,1,6,3] => [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [4,2,5,3,6,1] => [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [4,3,5,2,6,1] => [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,3,6,4,1,2] => [3,2,6,5,1,4] => 3 = 1 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [5,3,6,2,1,4] => [4,3,6,2,1,5] => 3 = 1 + 2
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 + 2
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => [2,1,3,5,4,6,7] => ? = 0 + 2
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [3,4,1,5,2,6,7] => [1,3,2,5,4,6,7] => 3 = 1 + 2
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 + 2
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [3,2,4,5,1,6,7] => [2,1,3,5,4,6,7] => ? = 0 + 2
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => [1,3,2,5,4,6,7] => 3 = 1 + 2
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [4,1,5,2,6,3,7] => [2,1,4,3,6,5,7] => ? = 0 + 2
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => [2,1,4,3,5,7,6] => ? = 0 + 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => [2,1,3,5,4,7,6] => ? = 0 + 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [4,5,1,6,2,7,3] => [1,3,2,5,4,7,6] => 3 = 1 + 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [4,2,1,5,3,6,7] => [3,2,1,5,4,6,7] => ? = 1 + 2
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [4,2,5,1,3,6,7] => [3,2,5,1,4,6,7] => ? = 0 + 2
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [4,5,2,1,3,6,7] => [3,5,2,1,4,6,7] => ? = 1 + 2
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [4,2,5,3,1,6,7] => [2,1,5,4,3,6,7] => ? = 0 + 2
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [4,2,5,1,6,3,7] => [2,1,4,3,6,5,7] => ? = 0 + 2
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [4,2,5,3,6,1,7] => [2,1,4,3,6,5,7] => ? = 0 + 2
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [4,2,5,1,6,7,3] => [2,1,4,3,5,7,6] => ? = 0 + 2
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [4,2,5,6,1,7,3] => [2,1,3,5,4,7,6] => ? = 0 + 2
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [4,5,2,6,1,7,3] => [1,3,2,5,4,7,6] => 3 = 1 + 2
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [4,2,5,3,6,7,1] => [2,1,4,3,5,7,6] => ? = 0 + 2
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [4,2,5,6,3,7,1] => [2,1,3,5,4,7,6] => ? = 0 + 2
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [4,5,2,6,3,7,1] => [1,3,2,5,4,7,6] => 3 = 1 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [4,3,2,5,1,6,7] => [3,2,1,5,4,6,7] => ? = 1 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [4,3,5,2,1,6,7] => [2,1,5,4,3,6,7] => ? = 0 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [4,3,5,2,6,1,7] => [2,1,4,3,6,5,7] => ? = 0 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => [2,1,4,3,5,7,6] => ? = 0 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [4,3,5,6,2,7,1] => [2,1,3,5,4,7,6] => ? = 0 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [4,5,3,6,2,7,1] => [1,3,2,5,4,7,6] => 3 = 1 + 2
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [5,2,1,6,3,7,4] => [3,2,1,5,4,7,6] => ? = 1 + 2
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [5,2,6,1,3,7,4] => [3,2,5,1,4,7,6] => ? = 0 + 2
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [5,6,2,1,3,7,4] => [3,5,2,1,4,7,6] => ? = 1 + 2
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [5,2,6,3,1,7,4] => [2,1,5,4,3,7,6] => ? = 0 + 2
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [5,2,6,1,7,3,4] => [3,2,5,1,7,4,6] => ? = 0 + 2
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [5,2,6,3,7,1,4] => [2,1,5,4,7,3,6] => ? = 0 + 2
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [5,2,6,7,3,1,4] => [2,1,5,7,4,3,6] => ? = 0 + 2
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [5,6,2,7,3,1,4] => [2,5,1,7,4,3,6] => ? = 1 + 2
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [5,2,6,3,7,4,1] => [2,1,4,3,7,6,5] => ? = 0 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [5,3,6,4,1,2,7] => [3,2,6,5,1,4,7] => ? = 0 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [5,3,1,6,4,7,2] => [3,2,1,5,4,7,6] => ? = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [5,3,6,1,4,7,2] => [3,2,5,1,4,7,6] => ? = 0 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [5,6,3,1,4,7,2] => [3,5,2,1,4,7,6] => ? = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [5,3,6,4,1,7,2] => [2,1,5,4,3,7,6] => ? = 0 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [5,3,6,1,7,4,2] => [2,1,4,3,7,6,5] => ? = 0 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [5,3,6,4,7,1,2] => [2,1,5,4,7,3,6] => ? = 0 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [5,3,6,7,4,1,2] => [2,1,4,7,6,3,5] => ? = 0 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [5,6,3,7,4,1,2] => [1,4,3,7,6,2,5] => 3 = 1 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [5,3,6,2,1,4,7] => [4,3,6,2,1,5,7] => ? = 0 + 2
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [5,3,2,6,1,7,4] => [3,2,1,5,4,7,6] => ? = 1 + 2
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [5,3,6,2,1,7,4] => [2,1,5,4,3,7,6] => ? = 0 + 2
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [5,3,6,2,7,1,4] => [2,1,5,4,7,3,6] => ? = 0 + 2
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [5,3,6,7,2,1,4] => [2,1,5,7,4,3,6] => ? = 0 + 2
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [5,6,3,7,2,1,4] => [1,5,4,7,3,2,6] => ? = 1 + 2
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [5,3,2,6,4,7,1] => [3,2,1,5,4,7,6] => ? = 1 + 2
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [5,3,6,2,4,7,1] => [3,2,5,1,4,7,6] => ? = 0 + 2
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [5,6,3,2,4,7,1] => [3,5,2,1,4,7,6] => ? = 1 + 2
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [5,3,6,4,2,7,1] => [2,1,5,4,3,7,6] => ? = 0 + 2
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [5,3,6,2,7,4,1] => [2,1,4,3,7,6,5] => ? = 0 + 2
[[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => [5,1,6,2,7,3,8,4] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,6,8],[4]]
=> [4,2,6,8,1,3,5,7] => [5,2,6,1,7,3,8,4] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,4,8],[6]]
=> [6,2,4,8,1,3,5,7] => [5,2,6,3,7,1,8,4] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,4,6],[8]]
=> [8,2,4,6,1,3,5,7] => [5,2,6,3,7,4,8,1] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,6],[4,8]]
=> [4,8,2,6,1,3,5,7] => [5,3,6,1,7,4,8,2] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,4],[6,8]]
=> [6,8,2,4,1,3,5,7] => [5,3,6,4,7,1,8,2] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,8],[4],[6]]
=> [6,4,2,8,1,3,5,7] => [5,3,6,2,7,1,8,4] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,6],[4],[8]]
=> [8,4,2,6,1,3,5,7] => [5,3,6,2,7,4,8,1] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2,4],[6],[8]]
=> [8,6,2,4,1,3,5,7] => [5,3,6,4,7,2,8,1] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
[[1,3,5,7],[2],[4],[6],[8]]
=> [8,6,4,2,1,3,5,7] => [5,4,6,3,7,2,8,1] => [2,1,4,3,6,5,8,7] => 4 = 2 + 2
Description
The number of big ascents of a permutation after prepending zero.
Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$.
It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents ([[St000021]]).
G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation [[Mp00086]].
[[St000646]] is the statistic without the border condition $\pi(0) = 0$.
Matching statistic: St000832
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000832: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000832: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,1,4,6,3,5] => 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,6,4,3] => 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 0
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 0
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => 1
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => ? = 0
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 0
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => ? = 0
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => ? = 0
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 0
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => ? = 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 0
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [1,4,3,5,2,6,7] => 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => ? = 0
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => ? = 0
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => ? = 0
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => ? = 0
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => ? = 0
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => 1
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => ? = 0
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 0
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => ? = 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => ? = 0
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => ? = 0
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ? = 0
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => ? = 0
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => ? = 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => ? = 0
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [1,4,3,5,2,7,6] => 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [2,1,5,4,3,7,6] => ? = 0
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [2,4,1,3,6,7,5] => ? = 0
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,1,4,6,3,7,5] => ? = 0
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 0
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [1,3,2,6,5,7,4] => 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [2,1,4,3,7,6,5] => ? = 0
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [2,1,5,6,4,3,7] => ? = 0
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,2,1,5,7,4,6] => ? = 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ? = 0
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [1,4,3,5,7,2,6] => 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,1,5,4,7,3,6] => ? = 0
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,1,3,7,6,5] => ? = 0
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,1,4,6,7,3,5] => ? = 0
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 0
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,3,2,6,7,5,4] => 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [2,5,4,1,6,3,7] => ? = 0
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,2,1,4,7,6] => ? = 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,4,1,3,7,6] => ? = 0
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ? = 0
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [2,1,6,5,3,7,4] => ? = 0
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [1,3,6,5,2,7,4] => 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,2,1,7,5,4,6] => ? = 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ? = 0
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [1,4,3,7,5,2,6] => 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [2,1,5,7,4,3,6] => ? = 0
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ? = 0
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [2,1,4,7,6,3,5] => ? = 0
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ? = 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ? = 0
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ? = 0
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ? = 0
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ? = 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [1,4,6,3,7,5,2] => 1
Description
The number of permutations obtained by reversing blocks of three consecutive numbers.
Matching statistic: St000623
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000623: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000623: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,1,4,6,3,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,6,4,3] => 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => 0 = 1 - 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => 0 = 1 - 1
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => 0 = 1 - 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => ? = 0 - 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 0 - 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [1,4,3,5,2,6,7] => 0 = 1 - 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => ? = 0 - 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => 0 = 1 - 1
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => ? = 0 - 1
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 0 - 1
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ? = 0 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => ? = 0 - 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => ? = 0 - 1
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [1,4,3,5,2,7,6] => 0 = 1 - 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [2,1,5,4,3,7,6] => ? = 0 - 1
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [2,4,1,3,6,7,5] => ? = 0 - 1
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,1,4,6,3,7,5] => ? = 0 - 1
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 0 - 1
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [1,3,2,6,5,7,4] => 0 = 1 - 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [2,1,4,3,7,6,5] => ? = 0 - 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [2,1,5,6,4,3,7] => ? = 0 - 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,2,1,5,7,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [1,4,3,5,7,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,1,5,4,7,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,1,3,7,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,1,4,6,7,3,5] => ? = 0 - 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 0 - 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,3,2,6,7,5,4] => 0 = 1 - 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [2,5,4,1,6,3,7] => ? = 0 - 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,2,1,4,7,6] => ? = 1 - 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,4,1,3,7,6] => ? = 0 - 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ? = 0 - 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [2,1,6,5,3,7,4] => ? = 0 - 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [1,3,6,5,2,7,4] => 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,2,1,7,5,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [1,4,3,7,5,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [2,1,5,7,4,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [2,1,4,7,6,3,5] => ? = 0 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ? = 1 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ? = 0 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ? = 0 - 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ? = 0 - 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ? = 1 - 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [1,4,6,3,7,5,2] => 0 = 1 - 1
Description
The number of occurrences of the pattern 52341 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to have a complete parabolic bundle structure [1].
Matching statistic: St000664
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000664: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,1,4,6,3,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,6,4,3] => 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => 0 = 1 - 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => 0 = 1 - 1
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => 0 = 1 - 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => ? = 0 - 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 0 - 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [1,4,3,5,2,6,7] => 0 = 1 - 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => ? = 0 - 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => 0 = 1 - 1
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => ? = 0 - 1
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 0 - 1
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ? = 0 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => ? = 0 - 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => ? = 0 - 1
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [1,4,3,5,2,7,6] => 0 = 1 - 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [2,1,5,4,3,7,6] => ? = 0 - 1
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [2,4,1,3,6,7,5] => ? = 0 - 1
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,1,4,6,3,7,5] => ? = 0 - 1
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 0 - 1
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [1,3,2,6,5,7,4] => 0 = 1 - 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [2,1,4,3,7,6,5] => ? = 0 - 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [2,1,5,6,4,3,7] => ? = 0 - 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,2,1,5,7,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [1,4,3,5,7,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,1,5,4,7,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,1,3,7,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,1,4,6,7,3,5] => ? = 0 - 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 0 - 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,3,2,6,7,5,4] => 0 = 1 - 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [2,5,4,1,6,3,7] => ? = 0 - 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,2,1,4,7,6] => ? = 1 - 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,4,1,3,7,6] => ? = 0 - 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ? = 0 - 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [2,1,6,5,3,7,4] => ? = 0 - 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [1,3,6,5,2,7,4] => 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,2,1,7,5,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [1,4,3,7,5,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [2,1,5,7,4,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [2,1,4,7,6,3,5] => ? = 0 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ? = 1 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ? = 0 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ? = 0 - 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ? = 0 - 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ? = 1 - 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [1,4,6,3,7,5,2] => 0 = 1 - 1
Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000666
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,1,4,6,3,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,6,4,3] => 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => 0 = 1 - 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => 0 = 1 - 1
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => 0 = 1 - 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => ? = 0 - 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 0 - 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [1,4,3,5,2,6,7] => 0 = 1 - 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => ? = 0 - 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => 0 = 1 - 1
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => ? = 0 - 1
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 0 - 1
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ? = 0 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => ? = 0 - 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => ? = 0 - 1
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [1,4,3,5,2,7,6] => 0 = 1 - 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [2,1,5,4,3,7,6] => ? = 0 - 1
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [2,4,1,3,6,7,5] => ? = 0 - 1
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,1,4,6,3,7,5] => ? = 0 - 1
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 0 - 1
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [1,3,2,6,5,7,4] => 0 = 1 - 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [2,1,4,3,7,6,5] => ? = 0 - 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [2,1,5,6,4,3,7] => ? = 0 - 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,2,1,5,7,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [1,4,3,5,7,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,1,5,4,7,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,1,3,7,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,1,4,6,7,3,5] => ? = 0 - 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 0 - 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,3,2,6,7,5,4] => 0 = 1 - 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [2,5,4,1,6,3,7] => ? = 0 - 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,2,1,4,7,6] => ? = 1 - 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,4,1,3,7,6] => ? = 0 - 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ? = 0 - 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [2,1,6,5,3,7,4] => ? = 0 - 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [1,3,6,5,2,7,4] => 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,2,1,7,5,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [1,4,3,7,5,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [2,1,5,7,4,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [2,1,4,7,6,3,5] => ? = 0 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ? = 1 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ? = 0 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ? = 0 - 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ? = 0 - 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ? = 1 - 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [1,4,6,3,7,5,2] => 0 = 1 - 1
Description
The number of right tethers of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right tether is a large ascent between two consecutive rafts of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St000801
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => 0 = 1 - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4,1,3,5,6] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [2,1,4,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4,1,3,6,5] => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,1,4,6,3,5] => 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => 0 = 1 - 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,6,4,3] => 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => 0 = 1 - 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [1,3,2,5,4,6,7] => 0 = 1 - 1
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => ? = 0 - 1
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => ? = 0 - 1
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [1,3,5,2,4,6,7] => 0 = 1 - 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => ? = 0 - 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [1,3,2,5,4,7,6] => 0 = 1 - 1
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 0 - 1
[[1,2,6,7],[3,5],[4]]
=> [4,3,5,1,2,6,7] => [1,4,3,5,2,6,7] => 0 = 1 - 1
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2,6],[4]]
=> [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => ? = 0 - 1
[[1,3,5,7],[2,4],[6]]
=> [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2,7],[4]]
=> [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => ? = 0 - 1
[[1,3,4,6],[2,7],[5]]
=> [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => ? = 0 - 1
[[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => [1,3,5,2,4,7,6] => 0 = 1 - 1
[[1,3,5,6],[2,4],[7]]
=> [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => ? = 0 - 1
[[1,3,4,6],[2,5],[7]]
=> [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => ? = 0 - 1
[[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => [1,3,2,5,7,4,6] => 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => ? = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => ? = 0 - 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => ? = 0 - 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => ? = 0 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => ? = 0 - 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 0 = 1 - 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => ? = 0 - 1
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [1,4,3,5,2,7,6] => 0 = 1 - 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [2,1,5,4,3,7,6] => ? = 0 - 1
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [2,4,1,3,6,7,5] => ? = 0 - 1
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,1,4,6,3,7,5] => ? = 0 - 1
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,1,3,6,5,7,4] => ? = 0 - 1
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [1,3,2,6,5,7,4] => 0 = 1 - 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [2,1,4,3,7,6,5] => ? = 0 - 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [2,1,5,6,4,3,7] => ? = 0 - 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,2,1,5,7,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [2,4,1,5,7,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [1,4,3,5,7,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,1,5,4,7,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,1,3,7,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,1,4,6,7,3,5] => ? = 0 - 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,1,3,6,7,5,4] => ? = 0 - 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,3,2,6,7,5,4] => 0 = 1 - 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [2,5,4,1,6,3,7] => ? = 0 - 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,2,1,4,7,6] => ? = 1 - 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,4,1,3,7,6] => ? = 0 - 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,4,6,1,3,7,5] => ? = 0 - 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [2,1,6,5,3,7,4] => ? = 0 - 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [1,3,6,5,2,7,4] => 0 = 1 - 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [3,2,1,7,5,4,6] => ? = 1 - 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [2,4,1,7,5,3,6] => ? = 0 - 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [1,4,3,7,5,2,6] => 0 = 1 - 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [2,1,5,7,4,3,6] => ? = 0 - 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [2,4,1,7,3,6,5] => ? = 0 - 1
[[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => [2,1,4,7,6,3,5] => ? = 0 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => ? = 1 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => ? = 0 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => ? = 0 - 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [2,5,4,1,7,6,3] => ? = 0 - 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [3,2,6,1,7,5,4] => ? = 1 - 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [1,4,6,3,7,5,2] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |312 in a permutation.
This is the number of occurrences of the pattern $(3,1,2)$, such that the letter matched by $3$ is the first entry of the permutation.
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000802The number of occurrences of the vincular pattern |321 in a permutation. St001130The number of two successive successions in a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001667The maximal size of a pair of weak twins for a permutation. St000028The number of stack-sorts needed to sort a permutation. St000054The first entry of the permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. 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. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000872The number of very big descents of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. 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. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000553The number of blocks of a graph.
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