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Matching statistic: St000716
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000716: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000716: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 21
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 21
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 21
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 14
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 21
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 14
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 14
Description
The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$.
For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.
Matching statistic: St001684
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(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 28%●distinct values known / distinct values provided: 20%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001684: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 28%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12 = 14 - 2
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 21 - 2
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,3,5],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,4,5],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,5,5],[3],[4]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 12 = 14 - 2
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 21 - 2
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 14 - 2
[[1,1,3,4],[2,2]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,1,4,4],[2,2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,1,3,4],[2,3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,1,4,4],[2,3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,1,4,4],[3,3]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,2,3,4],[2,3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,2,4,4],[2,3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,2,4,4],[3,3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[2,2,4,4],[3,3]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,1,1,4],[2],[3]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? = 14 - 2
[[1,1,2,4],[2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,1,3,4],[2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,1,4,4],[2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,2,2,4],[2],[3]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? = 14 - 2
[[1,2,3,4],[2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,2,4,4],[2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,3,3,4],[2],[3]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? = 14 - 2
[[1,3,4,4],[2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,4,4,4],[2],[3]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? = 14 - 2
[[1,1,3],[2,2],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,1,4],[2,2],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,1,4],[2,2],[4]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,1,3],[2,3],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,1,4],[2,3],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,1,4],[2,3],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,1,4],[2,4],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,1,4],[3,3],[4]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,2,3],[2,3],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,2,4],[2,3],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,2,4],[2,3],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,2,4],[2,4],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[1,2,4],[3,3],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 21 - 2
[[1,3,4],[2,4],[3]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
[[2,2,4],[3,3],[4]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 21 - 2
[[1,1,1],[2],[3],[4]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? = 14 - 2
[[1,1,2],[2],[3],[4]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 14 - 2
Description
The reduced word complexity of a permutation.
For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$.
For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$.
This statistic appears in [1, Question 6.1].
Matching statistic: St001870
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001870: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1],[2],[3],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[2],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[2],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 12
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2 = 14 - 12
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 12
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 2 = 14 - 12
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => 2 = 14 - 12
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 2 = 14 - 12
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 12
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [4,3,5,2,-6,1] => ? = 21 - 12
[[1,6],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,6],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,6],[2],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[2,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[2,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[2,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[3,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 2 = 14 - 12
[[1],[2],[3],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[2],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[2],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[2],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[2],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[2],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[3],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 2 = 14 - 12
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [3,1,-5,2,4] => ? = 21 - 12
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 12
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 12
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 12
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2 = 14 - 12
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2 = 14 - 12
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 12
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2 = 14 - 12
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2 = 14 - 12
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 12
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 2 = 14 - 12
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 2 = 14 - 12
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [3,4,1,-5,2] => ? = 21 - 12
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 12
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 12
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 12
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 12
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 12
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 12
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 12
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[2,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 12
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [5,4,1,2,3,6] => [3,4,-6,1,2,5] => ? = 14 - 12
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [5,3,1,2,4,6] => [2,4,-6,1,3,5] => ? = 14 - 12
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 12
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 12
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => [1,4,-6,2,3,5] => ? = 14 - 12
[[1,2,3,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 12
[[1,2,4,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 12
[[1,3,3,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 12
[[1,3,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 12
[[1,4,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 12
[[1,1,3],[2,2],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 12
[[1,1,4],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 14 - 12
[[1,1,4],[2,2],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 12
[[1,1,3],[2,3],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 12
[[1,1,4],[2,3],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 12
[[1,1,4],[2,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 12
[[1,1,4],[2,4],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 12
[[1,1,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 12
[[1,2,3],[2,3],[4]]
=> [6,2,4,1,3,5] => [6,2,4,1,3,5] => [3,5,1,-6,2,4] => ? = 21 - 12
[[1,2,4],[2,3],[3]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => [1,4,3,-6,2,5] => ? = 14 - 12
[[1,2,4],[2,3],[4]]
=> [5,2,4,1,3,6] => [5,2,4,1,3,6] => [3,4,1,-6,2,5] => ? = 21 - 12
[[1,2,4],[2,4],[3]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => [1,4,3,-6,2,5] => ? = 14 - 12
[[1,2,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 12
[[1,3,4],[2,4],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4,6] => [1,4,2,-6,3,5] => ? = 14 - 12
Description
The number of positive entries followed by a negative entry in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.
Matching statistic: St001895
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001895: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1],[2],[3],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[2],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 13
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 13
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 1 = 14 - 13
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => 1 = 14 - 13
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 1 = 14 - 13
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [4,3,5,2,-6,1] => ? = 21 - 13
[[1,6],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[2],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[3,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1],[2],[3],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[2],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[2],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[3],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [3,1,-5,2,4] => ? = 21 - 13
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 13
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 13
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 13
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 13
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 13
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1 = 14 - 13
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [3,4,1,-5,2] => ? = 21 - 13
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 13
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 13
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 13
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 13
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 13
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 13
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 13
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[2,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [5,4,1,2,3,6] => [3,4,-6,1,2,5] => ? = 14 - 13
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [5,3,1,2,4,6] => [2,4,-6,1,3,5] => ? = 14 - 13
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 13
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 13
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => [1,4,-6,2,3,5] => ? = 14 - 13
[[1,2,3,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 13
[[1,2,4,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 13
[[1,3,3,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 13
[[1,3,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 13
[[1,4,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 13
[[1,1,3],[2,2],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 13
[[1,1,4],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 14 - 13
[[1,1,4],[2,2],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,1,3],[2,3],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 13
[[1,1,4],[2,3],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 13
[[1,1,4],[2,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,1,4],[2,4],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 13
[[1,1,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,2,3],[2,3],[4]]
=> [6,2,4,1,3,5] => [6,2,4,1,3,5] => [3,5,1,-6,2,4] => ? = 21 - 13
[[1,2,4],[2,3],[3]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => [1,4,3,-6,2,5] => ? = 14 - 13
[[1,2,4],[2,3],[4]]
=> [5,2,4,1,3,6] => [5,2,4,1,3,6] => [3,4,1,-6,2,5] => ? = 21 - 13
[[1,2,4],[2,4],[3]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => [1,4,3,-6,2,5] => ? = 14 - 13
[[1,2,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,3,4],[2,4],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4,6] => [1,4,2,-6,3,5] => ? = 14 - 13
Description
The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
Matching statistic: St001868
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1],[2],[3],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[2],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[2],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 14
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 0 = 14 - 14
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 14
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 0 = 14 - 14
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => 0 = 14 - 14
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 0 = 14 - 14
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 14
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [4,3,5,2,-6,1] => ? = 21 - 14
[[1,6],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,6],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,6],[2],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[2,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[2,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[2,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[3,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 0 = 14 - 14
[[1],[2],[3],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[2],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[2],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[2],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[2],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[2],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[3],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 0 = 14 - 14
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [3,1,-5,2,4] => ? = 21 - 14
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 14
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 14
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 14
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 0 = 14 - 14
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 0 = 14 - 14
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 14
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 0 = 14 - 14
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 0 = 14 - 14
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => ? = 14 - 14
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 0 = 14 - 14
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 0 = 14 - 14
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [3,4,1,-5,2] => ? = 21 - 14
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 14
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 14
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 14
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 14
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => ? = 14 - 14
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 14
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 14
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[2,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 14
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [5,4,1,2,3,6] => [3,4,-6,1,2,5] => ? = 14 - 14
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [5,3,1,2,4,6] => [2,4,-6,1,3,5] => ? = 14 - 14
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 14
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 14
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => [1,4,-6,2,3,5] => ? = 14 - 14
[[1,2,3,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 14
[[1,2,4,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 14
[[1,3,3,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 14
[[1,3,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 14
[[1,4,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 14
[[1,1,3],[2,2],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 14
[[1,1,4],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 14 - 14
[[1,1,4],[2,2],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 14
[[1,1,3],[2,3],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 14
[[1,1,4],[2,3],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 14
[[1,1,4],[2,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 14
[[1,1,4],[2,4],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 14
[[1,1,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 14
[[1,2,3],[2,3],[4]]
=> [6,2,4,1,3,5] => [6,2,4,1,3,5] => [3,5,1,-6,2,4] => ? = 21 - 14
[[1,2,4],[2,3],[3]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => [1,4,3,-6,2,5] => ? = 14 - 14
[[1,2,4],[2,3],[4]]
=> [5,2,4,1,3,6] => [5,2,4,1,3,6] => [3,4,1,-6,2,5] => ? = 21 - 14
[[1,2,4],[2,4],[3]]
=> [4,2,5,1,3,6] => [4,2,5,1,3,6] => [1,4,3,-6,2,5] => ? = 14 - 14
[[1,2,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 14
[[1,3,4],[2,4],[3]]
=> [3,2,5,1,4,6] => [3,2,5,1,4,6] => [1,4,2,-6,3,5] => ? = 14 - 14
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Matching statistic: St001771
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001771: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 27%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1,5],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,5],[2],[4]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,5],[3],[4]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[2,5],[3],[4]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1],[2],[3],[5]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[2],[4],[5]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[3],[4],[5]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[2],[3],[4],[5]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 14 - 14
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 14 - 14
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 14 - 14
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [1,4,3,5,2] => [1,4,3,5,2] => 0 = 14 - 14
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [1,4,2,5,3] => [1,4,2,5,3] => 0 = 14 - 14
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 14 - 14
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => ? = 21 - 14
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [4,5,2,3,1,6] => [4,5,2,3,1,6] => ? = 21 - 14
[[1,6],[2],[3]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,6],[2],[4]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,6],[2],[5]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,6],[3],[4]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,6],[3],[5]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1,6],[4],[5]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[2,6],[3],[4]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[2,6],[3],[5]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[2,6],[4],[5]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[3,6],[4],[5]]
=> [3,2,1,4] => [1,2,4,3] => [1,2,4,3] => 0 = 14 - 14
[[1],[2],[3],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[2],[4],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[2],[5],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[3],[4],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[3],[5],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1],[4],[5],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[2],[3],[4],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[2],[3],[5],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[2],[4],[5],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[3],[4],[5],[6]]
=> [4,3,2,1] => [1,3,2,4] => [1,3,2,4] => 0 = 14 - 14
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 21 - 14
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 14 - 14
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 14 - 14
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 14 - 14
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 14 - 14
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 14 - 14
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 14 - 14
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 14 - 14
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 14 - 14
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 14 - 14
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => 0 = 14 - 14
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 14 - 14
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 21 - 14
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 14
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 14 - 14
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 14 - 14
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 14 - 14
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 14 - 14
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 14 - 14
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [6,3,2,4,5,1] => [6,3,2,4,5,1] => ? = 21 - 14
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [6,3,2,4,5,1] => [6,3,2,4,5,1] => ? = 21 - 14
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[2,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 21 - 14
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [4,3,6,2,5,1] => [4,3,6,2,5,1] => ? = 14 - 14
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [4,2,6,3,5,1] => [4,2,6,3,5,1] => ? = 14 - 14
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [3,2,6,4,5,1] => [3,2,6,4,5,1] => ? = 14 - 14
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [3,2,6,4,5,1] => [3,2,6,4,5,1] => ? = 14 - 14
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => ? = 14 - 14
[[1,2,3,4],[2],[3]]
=> [4,2,1,3,5,6] => [1,3,6,2,4,5] => [1,3,6,2,4,5] => ? = 14 - 14
[[1,2,4,4],[2],[3]]
=> [4,2,1,3,5,6] => [1,3,6,2,4,5] => [1,3,6,2,4,5] => ? = 14 - 14
[[1,3,3,4],[2],[3]]
=> [3,2,1,4,5,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 14 - 14
[[1,3,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 14 - 14
[[1,4,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 14 - 14
[[1,1,3],[2,2],[4]]
=> [6,3,4,1,2,5] => [5,2,3,6,4,1] => [5,2,3,6,4,1] => ? = 21 - 14
[[1,1,4],[2,2],[3]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => ? = 14 - 14
[[1,1,4],[2,2],[4]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => ? = 21 - 14
[[1,1,3],[2,3],[4]]
=> [6,3,4,1,2,5] => [5,2,3,6,4,1] => [5,2,3,6,4,1] => ? = 21 - 14
[[1,1,4],[2,3],[3]]
=> [4,3,5,1,2,6] => [3,2,4,6,5,1] => [3,2,4,6,5,1] => ? = 14 - 14
[[1,1,4],[2,3],[4]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => ? = 21 - 14
[[1,1,4],[2,4],[3]]
=> [4,3,5,1,2,6] => [3,2,4,6,5,1] => [3,2,4,6,5,1] => ? = 14 - 14
[[1,1,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => ? = 21 - 14
[[1,2,3],[2,3],[4]]
=> [6,2,4,1,3,5] => [5,6,3,2,4,1] => [5,6,3,2,4,1] => ? = 21 - 14
[[1,2,4],[2,3],[3]]
=> [4,2,5,1,3,6] => [3,6,4,2,5,1] => [3,6,4,2,5,1] => ? = 14 - 14
[[1,2,4],[2,3],[4]]
=> [5,2,4,1,3,6] => [4,6,3,2,5,1] => [4,6,3,2,5,1] => ? = 21 - 14
[[1,2,4],[2,4],[3]]
=> [4,2,5,1,3,6] => [3,6,4,2,5,1] => [3,6,4,2,5,1] => ? = 14 - 14
[[1,2,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [4,2,3,6,5,1] => [4,2,3,6,5,1] => ? = 21 - 14
Description
The number of occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Matching statistic: St001429
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00281: Signed permutations —rowmotion⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1],[2],[3],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[2],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1 = 14 - 13
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => 1 = 14 - 13
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 1 = 14 - 13
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => ? = 14 - 13
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 1 = 14 - 13
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [4,3,5,2,-6,1] => ? = 21 - 13
[[1,6],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[2],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,6],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,6],[3],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[2,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[3,6],[4],[5]]
=> [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1 = 14 - 13
[[1],[2],[3],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[2],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[2],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[2],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[3],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1 = 14 - 13
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,1,3,5] => [3,1,-5,2,4] => ? = 21 - 13
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1 = 14 - 13
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1 = 14 - 13
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1 = 14 - 13
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1 = 14 - 13
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1 = 14 - 13
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1 = 14 - 13
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => ? = 14 - 13
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [3,4,1,-5,2] => ? = 21 - 13
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,5,1,4] => [1,4,2,-5,3] => 1 = 14 - 13
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => 1 = 14 - 13
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => 1 = 14 - 13
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => 1 = 14 - 13
[[1,2],[2],[3],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 1 = 14 - 13
[[1,3],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => ? = 14 - 13
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => ? = 14 - 13
[[1,4],[2],[4],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => ? = 14 - 13
[[1,4],[3],[4],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => ? = 14 - 13
[[2,4],[3],[4],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => ? = 14 - 13
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 13
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,-6,2,4,5] => ? = 21 - 13
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[2,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [3,2,-6,1,4,5] => ? = 21 - 13
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [5,4,1,2,3,6] => [3,4,-6,1,2,5] => ? = 14 - 13
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [5,3,1,2,4,6] => [2,4,-6,1,3,5] => ? = 14 - 13
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 13
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [2,3,-6,1,4,5] => ? = 14 - 13
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => [1,4,-6,2,3,5] => ? = 14 - 13
[[1,2,3,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 13
[[1,2,4,4],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [1,3,-6,2,4,5] => ? = 14 - 13
[[1,3,3,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 13
[[1,3,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 13
[[1,4,4,4],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,2,-6,3,4,5] => ? = 14 - 13
[[1,1,3],[2,2],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 13
[[1,1,4],[2,2],[3]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 14 - 13
[[1,1,4],[2,2],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,1,3],[2,3],[4]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [3,5,2,-6,1,4] => ? = 21 - 13
[[1,1,4],[2,3],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 13
[[1,1,4],[2,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,1,4],[2,4],[3]]
=> [4,3,5,1,2,6] => [4,3,5,1,2,6] => [2,4,3,-6,1,5] => ? = 14 - 13
[[1,1,4],[3,3],[4]]
=> [5,3,4,1,2,6] => [5,3,4,1,2,6] => [3,4,2,-6,1,5] => ? = 21 - 13
[[1,2,3],[2,3],[4]]
=> [6,2,4,1,3,5] => [6,2,4,1,3,5] => [3,5,1,-6,2,4] => ? = 21 - 13
Description
The number of negative entries in a signed permutation.
Matching statistic: St001866
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001866: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1,5],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,5],[2],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,5],[3],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[2,5],[3],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1],[2],[3],[5]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[2],[4],[5]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[3],[4],[5]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[2],[3],[4],[5]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 14 - 14
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 14 - 14
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 14 - 14
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0 = 14 - 14
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 14 - 14
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 14 - 14
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [6,3,5,1,4,2] => [6,3,5,1,4,2] => ? = 21 - 14
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,3,6,1,4,2] => [5,3,6,1,4,2] => ? = 21 - 14
[[1,6],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,6],[2],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,6],[2],[5]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,6],[3],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,6],[3],[5]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1,6],[4],[5]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[2,6],[3],[4]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[2,6],[3],[5]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[2,6],[4],[5]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[3,6],[4],[5]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 14 - 14
[[1],[2],[3],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[2],[4],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[2],[5],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[3],[4],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[3],[5],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1],[4],[5],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[2],[3],[4],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[2],[3],[5],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[2],[4],[5],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[3],[4],[5],[6]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 14 - 14
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 21 - 14
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 14 - 14
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 14 - 14
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 14 - 14
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 14 - 14
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 14 - 14
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 14 - 14
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 14 - 14
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 14 - 14
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 14 - 14
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 14 - 14
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 14 - 14
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 21 - 14
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 14 - 14
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 14 - 14
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 14 - 14
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 14 - 14
[[1,2],[2],[3],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0 = 14 - 14
[[1,2],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0 = 14 - 14
[[1,3],[2],[3],[5]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 14 - 14
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 14 - 14
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0 = 14 - 14
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 14 - 14
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 14 - 14
[[1,5],[2],[3],[5]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 14 - 14
[[1,4],[2],[4],[5]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 14 - 14
[[1,5],[2],[4],[5]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 14 - 14
[[1,3],[3],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0 = 14 - 14
[[1,4],[3],[4],[5]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 14 - 14
[[1,5],[3],[4],[5]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 14 - 14
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0 = 14 - 14
[[2,3],[3],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0 = 14 - 14
[[2,4],[3],[4],[5]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 14 - 14
[[2,5],[3],[4],[5]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 14 - 14
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 14 - 14
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => ? = 21 - 14
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => ? = 21 - 14
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[2,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 21 - 14
[[1,1,1,4],[2],[3]]
=> [5,4,1,2,3,6] => [6,1,4,5,3,2] => [6,1,4,5,3,2] => ? = 14 - 14
[[1,1,2,4],[2],[3]]
=> [5,3,1,2,4,6] => [6,1,3,5,4,2] => [6,1,3,5,4,2] => ? = 14 - 14
[[1,1,3,4],[2],[3]]
=> [4,3,1,2,5,6] => [6,5,1,3,4,2] => [6,5,1,3,4,2] => ? = 14 - 14
[[1,1,4,4],[2],[3]]
=> [4,3,1,2,5,6] => [6,5,1,3,4,2] => [6,5,1,3,4,2] => ? = 14 - 14
[[1,2,2,4],[2],[3]]
=> [5,2,1,3,4,6] => [6,1,2,5,4,3] => [6,1,2,5,4,3] => ? = 14 - 14
[[1,2,3,4],[2],[3]]
=> [4,2,1,3,5,6] => [6,5,1,2,4,3] => [6,5,1,2,4,3] => ? = 14 - 14
Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
* $-i < -j < -\pi(j) < -\pi(i)$, or
* $-i < j \leq \pi(j) < -\pi(i)$, or
* $i < j \leq \pi(j) < \pi(i)$.
Matching statistic: St001892
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001892: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1],[2],[3],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[2],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [3,5,6,1,4,2] => [3,5,6,1,4,2] => ? = 21 - 10
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 21 - 10
[[1,6],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[2],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[2],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[3],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[4],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,6],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,6],[3],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,6],[4],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[3,6],[4],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1],[2],[3],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[2],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[2],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[3],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 21 - 10
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 21 - 10
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 14 - 10
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,2],[2],[3],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,2],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,3],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,5],[2],[3],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,4],[2],[4],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,5],[2],[4],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,3],[3],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,4],[3],[4],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,5],[3],[4],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[2,3],[3],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[2,4],[3],[4],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[2,5],[3],[4],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 14 - 10
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,5,6,1,3] => [2,4,5,6,1,3] => ? = 21 - 10
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,5,6,1,3] => [2,4,5,6,1,3] => ? = 21 - 10
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
Description
The flag excedance statistic of a signed permutation.
This is the number of negative entries plus twice the number of excedances of the signed permutation. That is,
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
where
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$
It has the same distribution as the flag descent statistic.
Matching statistic: St001894
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 20%
Values
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1,5],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,5],[2],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,5],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,5],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1],[2],[3],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[2],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[3],[4],[5]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1,1,4],[2],[3]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,3,4],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,4,4],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,1],[2],[3],[4]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,2],[2],[3],[4]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,3],[2],[3],[4]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,4],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,3],[2,2],[3]]
=> [5,3,4,1,2,6] => [3,5,6,1,4,2] => [3,5,6,1,4,2] => ? = 21 - 10
[[1,1],[2,2],[3,3]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 21 - 10
[[1,6],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[2],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[2],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[3],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1,6],[4],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,6],[3],[4]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,6],[3],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[2,6],[4],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[3,6],[4],[5]]
=> [3,2,1,4] => [3,4,2,1] => [3,4,2,1] => 4 = 14 - 10
[[1],[2],[3],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[2],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[2],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[3],[4],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[3],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[2],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[3],[4],[5],[6]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 14 - 10
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 21 - 10
[[1,1,5],[2],[3]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,1,5],[2],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,1,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,2,5],[2],[3]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,2,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,3,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,5,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,4,5],[2],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,5,5],[2],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,3,5],[3],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[1,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[2,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 4 = 14 - 10
[[2,3,5],[3],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 14 - 10
[[2,4,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[2,5,5],[3],[4]]
=> [3,2,1,4,5] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 14 - 10
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 21 - 10
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 14 - 10
[[1,1],[2],[3],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,1],[2],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,1],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,2],[2],[3],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,2],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,3],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,5],[2],[3],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,4],[2],[4],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,5],[2],[4],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1,3],[3],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[1,4],[3],[4],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[1,5],[3],[4],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[2,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 14 - 10
[[2,3],[3],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 14 - 10
[[2,4],[3],[4],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 14 - 10
[[2,5],[3],[4],[5]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 14 - 10
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 14 - 10
[[1,1,3,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,4,4],[2,2]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,3,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,4,4],[2,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,1,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
[[1,2,3,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,5,6,1,3] => [2,4,5,6,1,3] => ? = 21 - 10
[[1,2,4,4],[2,3]]
=> [2,4,1,3,5,6] => [2,4,5,6,1,3] => [2,4,5,6,1,3] => ? = 21 - 10
[[1,2,4,4],[3,3]]
=> [3,4,1,2,5,6] => [3,4,5,6,1,2] => [3,4,5,6,1,2] => ? = 21 - 10
Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001817The number of flag weak exceedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001852The size of the conjugacy class of the signed permutation. St000165The sum of the entries of a parking function. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000540The sum of the entries of a parking function minus its length. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000135The number of lucky cars of the parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001822The number of alignments of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000070The number of antichains in a poset. St001779The order of promotion on the set of linear extensions of a poset. St001858The number of covering elements of a signed permutation in absolute order. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001865The number of alignments of a signed permutation. St000016The number of attacking pairs of a standard tableau. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001782The order of rowmotion on the set of order ideals of a poset. St001854The size of the left Kazhdan-Lusztig cell, St001848The atomic length of a signed permutation. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000641The number of non-empty boolean intervals in a poset. St000656The number of cuts of a poset. St001819The flag Denert index of a signed permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001717The largest size of an interval in a poset. St001721The degree of a binary word. St000017The number of inversions of a standard tableau. St000072The number of circled entries. St000073The number of boxed entries. St000077The number of boxed and circled entries. St000189The number of elements in the poset. St000526The number of posets with combinatorially isomorphic order polytopes. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001343The dimension of the reduced incidence algebra of a poset. St001433The flag major index of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001718The number of non-empty open intervals in a poset. St001902The number of potential covers of a poset. St000044The number of vertices of the unicellular map given by a perfect matching. St000639The number of relations in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000912The number of maximal antichains in a poset. St001268The size of the largest ordinal summand in the poset. St001397Number of pairs of incomparable elements in a finite poset. St001768The number of reduced words of a signed permutation. St001821The sorting index of a signed permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000528The height of a poset. St000744The length of the path to the largest entry in a standard Young tableau. St000849The number of 1/3-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000958The number of Bruhat factorizations of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001893The flag descent of a signed permutation. St000080The rank of the poset. St000717The number of ordinal summands of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000942The number of critical left to right maxima of the parking functions. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001209The pmaj statistic of a parking function. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St001399The distinguishing number of a poset. St001423The number of distinct cubes in a binary word. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001853The size of the two-sided Kazhdan-Lusztig cell, St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000068The number of minimal elements in a poset. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000850The number of 1/2-balanced pairs in a poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001396Number of triples of incomparable elements in a finite poset. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001770The number of facets of a certain subword complex associated with the signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001884The number of borders of a binary word. St001903The number of fixed points of a parking function. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000295The length of the border of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001301The first Betti number of the order complex associated with the poset. St001371The length of the longest Yamanouchi prefix of a binary word. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001730The number of times the path corresponding to a binary word crosses the base line. St001851The number of Hecke atoms of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001664The number of non-isomorphic subposets of a poset. St001885The number of binary words with the same proper border set. St001709The number of homomorphisms to the three element chain of a poset. St001815The number of order preserving surjections from a poset to a total order. St001813The product of the sizes of the principal order filters in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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