A study of unimodality of some combinatorial sequences and polynomials

TL;DR

This paper explores the property of unimodality in combinatorial sequences, focusing on classical results and the Gaussian polynomial, including attempts to prove its unimodality through injective methods.

Contribution

It provides a concise overview of unimodality in combinatorics, examines the Gaussian polynomial's unimodality, and discusses the challenges in establishing an injective proof.

Findings

Review of classical unimodality results

Analysis of Gaussian polynomial unimodality

Identification of challenges in injective proofs

Abstract

In this article, we present a short, non-exhaustive study of an important and well-known property of combinatorial sequences - unimodality. We shall have a look at a sample of classical results on unimodality and related properties, and then proceed to understand the unimodality of the Gaussian polynomial in more detail. We will look at an outline of O'Hara's proof of the unimodality of the Gaussian polynomial. In order to grasp the challenge of the problem of obtaining an injective proof of the unimodality of the Gaussian polynomial (which is still an open question), we make several attempts and understand where these attempts fail.