Evacuation from StandardTableau

Schützenberger's evacuation (involution): remove the entry 1, slide the rest in, and give the cell that empties the largest label, until nothing is left.

StandardTableau → StandardTableauinvolution

FindStat's Mp00085. Computed as the insertion tableau of the reading word reversed and complemented; the same shape, and an involution.

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

Live

Explore

Each element of StandardTableau at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on StandardTableau. Its distribution over the images of StandardTableau at each size is computed, and compared with the 25 statistics on StandardTableau: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic RowOfLargestEntryLastEntryInFirstRowLastEntryOnMainDiagonalFirstEntryInLastRowAdjacentSuccessorCountPathLengthToLargestEntryFirstColumnInitialSequenceLengthModularMajorIndexNaturalDescentCountNaturalMajorIndexNaturalComajorIndexChargeCochargeWeightedSizeAttackingPairCountAttackingInversionCountShynarInversionCountAscentCountIncludingLargestEntryShiftedNaturalComajorIndexCylindricalTableauOverlapMinimalCylindricalInnerHeightSmallestCylindricShiftMajorIndexMinusWeightedSizeComajorIndexMinusWeightedSizeHaglundStevensInversionCount Sizes 1 to

FindStat Mp00085

Schützenberger involution

Matched by value: our map agrees with FindStat's image on each of the 119 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

BijectionGraded mapInvolutionas FindStat records them, checked by FindStat's contributors.

Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.

Object	Image
[[1]]	[[1]]
[[1,2]]	[[1,2]]
[[1,2,3]]	[[1,2,3]]
[[1],[2]]	[[1],[2]]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00085 has the rest.