Compositions inside a rectangle and unimodality

TL;DR

This paper proves algebraically that the sequence counting compositions fitting inside a rectangle is unimodal, addressing a conjecture and discussing the open problem of a combinatorial proof.

Contribution

It provides a simple algebraic proof for the unimodality of compositions within a rectangle, confirming a conjecture by Vatter.

Findings

Sequence c^{k,l}(n) is unimodal.

Algebraic proof established for the conjecture.

Open problem of combinatorial proof discussed.

Abstract

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.