Identifier
Values
[[1]] => [1] => [1] => [-1] => 1
[[1,2]] => [1,2] => [1,2] => [-2,1] => 1
[[1],[2]] => [2,1] => [2,1] => [1,-2] => 1
[[1,2,3]] => [1,2,3] => [1,2,3] => [-3,1,2] => 1
[[1,3],[2]] => [2,1,3] => [2,1,3] => [1,-3,2] => 1
[[1,2],[3]] => [3,1,2] => [3,1,2] => [2,-3,1] => 1
[[1],[2],[3]] => [3,2,1] => [3,2,1] => [1,2,-3] => 1
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => [-4,1,2,3] => 1
[[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => [1,-4,2,3] => 1
[[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => [2,-4,1,3] => 1
[[1,2,3],[4]] => [4,1,2,3] => [4,1,2,3] => [3,-4,1,2] => 1
[[1,3],[2,4]] => [2,4,1,3] => [2,4,1,3] => [3,1,-4,2] => 2
[[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => [3,2,-4,1] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => [1,2,-4,3] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [4,2,1,3] => [1,3,-4,2] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [4,3,1,2] => [2,3,-4,1] => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => 1
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => [-5,1,2,3,4] => 1
[[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => [1,-5,2,3,4] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,1,2,4,5] => [2,-5,1,3,4] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [4,1,2,3,5] => [3,-5,1,2,4] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [5,1,2,3,4] => [4,-5,1,2,3] => 1
[[1,3,5],[2,4]] => [2,4,1,3,5] => [2,4,1,3,5] => [3,1,-5,2,4] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2,5] => [3,2,-5,1,4] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [2,5,1,3,4] => [4,1,-5,2,3] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [3,5,1,2,4] => [4,2,-5,1,3] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [4,5,1,2,3] => [4,3,-5,1,2] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => [1,2,-5,3,4] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [4,2,1,3,5] => [1,3,-5,2,4] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [4,3,1,2,5] => [2,3,-5,1,4] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [5,2,1,3,4] => [1,4,-5,2,3] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [5,3,1,2,4] => [2,4,-5,1,3] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [5,4,1,2,3] => [3,4,-5,1,2] => 1
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,2,5,1,4] => [1,4,2,-5,3] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => [1,4,3,-5,2] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,3,5,1,2] => [2,4,3,-5,1] => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [5,2,4,1,3] => [3,4,1,-5,2] => 2
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [5,3,4,1,2] => [3,4,2,-5,1] => 2
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [5,3,2,1,4] => [1,2,4,-5,3] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [5,4,2,1,3] => [1,3,4,-5,2] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [5,4,3,1,2] => [2,3,4,-5,1] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,-5] => 1
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Description
The number of descents of a signed permutation.
A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.
Map
rowmotion
Description
The rowmotion of a signed permutation with respect to the sorting order.
The sorting order on signed permutations (with respect to the Coxeter element $-n, 1, 2,\dots, n-1$) is defined in [1].
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.