FlagExcedance on SignedPermutation

Twice the excedances plus the negative entries.

On SignedPermutation

FindStat's St001892. An excedance is an entry p(i) > i.

Called as CombinatorialStat(x, "FlagExcedance"). The head's page has its examples.

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Distribution

Over SignedPermutations at each size, tallied here in your browser; sizes past 6 are not offered.

Sizes 1 to

FindStat St001892

The flag excedance statistic of a signed permutation.

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

$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|$$$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|$$

Object	Value
[1]	0
[-1]	1
[1,2]	0
[2,1]	2
[-1,2]	1
[-2,1]	1
[1,-2]	1
[2,-1]	3

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001892 has the rest.