Compositions inside a rectangle and unimodality
Abstract
Let ck,l(n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k-by-l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the sequence ck,l(0), ck,l(1), ..., ck,l(kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open.
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