Identifier
Values
[[1]] => [1] => [1] => [1] => 0
[[1,2]] => [1,2] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [2,1] => [-2,1] => 1
[[1,2,3]] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [2,1,3] => [-2,1,3] => 1
[[1,2],[3]] => [3,1,2] => [3,1,2] => [3,1,2] => 2
[[1],[2],[3]] => [3,2,1] => [3,2,1] => [-2,-3,1] => 2
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => 1
[[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2
[[1,2,3],[4]] => [4,1,2,3] => [4,1,2,3] => [1,4,2,3] => 2
[[1,3],[2,4]] => [2,4,1,3] => [2,4,1,3] => [-2,4,1,3] => 3
[[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 4
[[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => 2
[[1,3],[2],[4]] => [4,2,1,3] => [4,2,1,3] => [-4,-2,1,3] => 2
[[1,2],[3],[4]] => [4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => 3
[[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => [-2,-3,-4,1] => 3
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,5],[4]] => [4,1,2,3,5] => [4,1,2,3,5] => [1,4,2,3,5] => 2
[[1,2,3,4],[5]] => [5,1,2,3,4] => [5,1,2,3,4] => [1,2,5,3,4] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [4,5,1,2,3] => [1,4,5,2,3] => 4
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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.
Map
Foata-Han
Description
The Foata-Han bijection for signed permutations.
This map sends the flag major index St001433The flag major index of a signed permutation. to the flag inversion number St001428The number of B-inversions of a signed permutation..
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.
Map
to signed permutation
Description
The signed permutation with all signs positive.