Garsia--Remmel $q$-rook numbers are not always unimodal

TL;DR

This paper demonstrates through an explicit example that Garsia--Remmel q-rook numbers for Ferrers boards can lack unimodality, answering a longstanding open question negatively.

Contribution

It provides the first explicit counterexample showing that Garsia--Remmel q-rook numbers are not always unimodal, resolving a question from 1986.

Findings

Garsia--Remmel q-rook numbers can be non-unimodal.

Counterexample disproves the conjecture of universal unimodality.

The result clarifies the behavior of q-rook numbers for Ferrers boards.

Abstract

We show by an explicit example that the Garsia--Remmel $q$-rook numbers of Ferrers boards do not all have unimodal sequences of coefficients. This resolves in the negative a question from 1986 by the aforementioned authors.