SmallestCylindricShift on StandardTableau
The least shift s at which each entry of the last row is smaller than the first row's entry s columns on.
Shifts below width - last row length cannot work, and a shift that works keeps working one column on, so the least one is that bound plus the shifts above it that fail (FindStat's St001993). The empty tableau gives 0.
Called as CombinatorialStat(x, "SmallestCylindricShift"). 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 St001993
The smallest shift such that the standard tableau is cylindric.
| Object | Value |
|---|---|
[[1]] | 1 |
[[1,2]] | 1 |
[[1,2,3]] | 1 |
[[1],[2]] | 1 |
[[1,2,3,4]] | 1 |
[[1,2],[3]] | 2 |
[[1,3],[2]] | 1 |
[[1,2,3,4,5]] | 1 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001993 has the rest.