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.