Garsia--Remmel $q$-rook numbers are not always unimodal
Combinatorics
2026-02-19 v3
Abstract
We show by an explicit example that the Garsia--Remmel -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.
Keywords
Cite
@article{arxiv.2410.19714,
title = {Garsia--Remmel $q$-rook numbers are not always unimodal},
author = {Joel Brewster Lewis and Alejandro H. Morales},
journal= {arXiv preprint arXiv:2410.19714},
year = {2026}
}
Comments
9 pages. v3: expanded Remark 6, added a table with more examples of q-rook numbers that are not unimodal. Ancillary files include python code to generate examples of q-rook numbers that are not unimodal