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CyclicCrossings on Permutation ​

Pairs (i, j) with i, p(j), p(i), j in cyclic order.

On Permutation

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

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Distribution

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FindStat St001537

The number of cyclic crossings of a permutation.

ObjectValue
[1]0
[1,2]0
[2,1]0
[1,2,3]0
[1,3,2]0
[2,1,3]0
[2,3,1]0
[3,1,2]0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001537 has the rest.