## The Dominance Lemma for Tabloids

If , and appears in a lower row than in the Young tabloid , then dominates . That is, swapping two entries of so as to move the lower number to a higher row moves the tabloid up in the dominance relations.

Let the composition sequences of and be and , respectively. For and we automatically have . For there is a difference between the two: the entry has been added in a different place. Let and be in rows and of , respectively. In , the entry is added to row , while in it’s been added to row . That is, is the same as with part increased by one and part decreased by one. Our assumption that is in a lower row than in is that . Therefore, since the lower row in is less than in , we find that . And we conclude that , as asserted.

[...] any such pair, the dominance lemma tells us that . That is, by “untwisting” the column inversion, we can move up the [...]

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[...] one thing at least will make our life simpler: it should be clear that we still have a dominance lemma for column dominance. To be explicit: if , and appears in a column to the right of in the column [...]

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[...] of course have a dominance lemma: if , occurs in a column to the left of in , and is obtained from by swapping these two [...]

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