Compositions inside a rectangle and unimodality
Abstract
Description
Let c^{k,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 c^{k,l}(0), c^{k,l}(1), ..., c^{k,l}(kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open.
9 pages, 1 figure, see related papers at http://www.math.msu.edu/~sagan
9 pages, 1 figure, see related papers at http://www.math.msu.edu/~sagan