ShiftedNaturalComajorIndex on StandardTableau
The sum of n - e over the natural descents e, plus the sum of (r - 1) times the length of row r.
The natural comajor index shifted by the weighted size of the shape (FindStat's St001697). A tableau of fewer than two entries has no natural descent and a weighted size of 0.
Called as CombinatorialStat(x, "ShiftedNaturalComajorIndex"). The head's page has its examples.
Live
Distribution
Over StandardTableaux at each size, tallied here in your browser; sizes past 6 are not offered.
FindStat St001697
The shifted natural comajor index of a standard Young tableau.
| Object | Value |
|---|---|
[[1]] | 0 |
[[1,2]] | 0 |
[[1,2,3]] | 0 |
[[1],[2]] | 1 |
[[1,2,3,4]] | 0 |
[[1,2],[3]] | 1 |
[[1,3],[2]] | 2 |
[[1,2,3,4,5]] | 0 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001697 has the rest.