TwoRowTableau from DyckPath

The standard tableau of shape (n, n) with the positions of the up steps in its first row and of the down steps in its second.

DyckPath → StandardTableau

FindStat's Mp00033. Injective; the same tableau Promotion and InversePromotion act on.

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

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Each element of DyckPath at a size, and its image, applied in your browser; sizes past 7 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on StandardTableau. Its distribution over the images of DyckPath at each size is computed, and compared with the 171 statistics on DyckPath: 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 Mp00033

to two-row standard tableau

Matched by value: our map agrees with FindStat's image on each of the 196 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

InjectionGraded mapas FindStat records them, checked by FindStat's contributors.

Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.

Object	Image
[1,0]	[[1],[2]]
[1,0,1,0]	[[1,3],[2,4]]
[1,1,0,0]	[[1,2],[3,4]]
[1,0,1,0,1,0]	[[1,3,5],[2,4,6]]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00033 has the rest.