AttackingInversionCount on StandardTableau
Attacking pairs of cells whose entries are out of order: the cell in the lower row, strictly east of the other, holds the larger entry.
Pairs in a row are in order. Across rows, the lower cell's entry exceeds the entry above it, which exceeds the entry of the cell it attacks, so every such pair is an inversion (FindStat's St000017). The empty tableau gives 0.
Called as CombinatorialStat(x, "AttackingInversionCount"). The head's page has its examples.
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Distribution
Over StandardTableaux at each size, tallied here in your browser; sizes past 6 are not offered.
FindStat St000017
The number of inversions of a standard tableau.
| Object | Value |
|---|---|
[[1]] | 0 |
[[1,2]] | 0 |
[[1,2,3]] | 0 |
[[1],[2]] | 0 |
[[1,2,3,4]] | 0 |
[[1,2],[3]] | 0 |
[[1,3],[2]] | 0 |
[[1,2,3,4,5]] | 0 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000017 has the rest.