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NormalizedLeafLabelSum on Permutation ​

The sum of the leaf labels of the increasing binary tree of p, less n.

On Permutation

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

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FindStat St000726

The normalized sum of the leaf labels of the increasing binary tree associated to a permutation.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000726 has the rest.