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DiagonalIndex on IntegerPartition ​

The sum of the contents j - i of the cells.

On IntegerPartition

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

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FindStat St000509

The diagonal index (content) of a partition.

ObjectValue
[2]1
[3]3
[4]6
[5]10
[6]15
[7]21
[8]28
[9]36

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000509 has the rest.