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

Joel Brewster Lewis, Alejandro H. Morales

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.

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

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.

Paper Structure

This paper contains 1 section, 3 theorems, 17 equations, 1 figure.

Table of Contents

  1. Acknowledgements

Key Result

Proposition 1

For every partition $\lambda = \langle \lambda_1, \ldots, \lambda_\ell\rangle$, $R_{\ell}(\lambda; q)$ is unimodal.

Figures (1)

  • Figure 1: The rook placement $\{(1, 5), (3, 1), (4, 3)\}$ on the board $B_{\langle 6, 5, 3, 3\rangle}$ has inversion number $5$. (Here we use matrix coordinates.)

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 3
  • Proposition 4
  • proof
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 2 more