Let and be partitions of the same size. The Kostka Number is the number of semistandard -tableau of type . It is easy to see that if then , where is the dominance order on partitions. I asked on MathOverflow if there was a short combinatorial proof of the converse. Matt Fayers posted an unexpectedly beautiful constructive answer. He has since put a proof of the stronger result that if then on his website.

The purpose of this post is to record a small simplification of Fayers’ original MathOverflow proof and to ask a question inspired by his stronger result.

#### Claim

Let have parts. If is maximal such that then there is a semistandard Young tableau of shape and content in which the entries equal to lie in rows numbered .

#### Proof of claim

We put an at the end of row of of the Young diagram of . Continuing inductively, we are left with partitions and obtained by deleting the boxes at the ends of row of and row of . Suppose, for a contradiction, that . Then

for some such that . (If then the final summand on the left is .) Since we have

Hence if and are the partitions obtained from and by removing the first parts of each then . But since , all the parts of are strictly less than the smallest part of , a contradiction.

If then we are finished with entries equal to . Otherwise since , the new value of is at least the old one (with equality unless ), as required for the claim.

#### Question

Is there a nice characterization of the triples such that and ?

The pairs such that have been characterized.