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Positivity properties of $q$-hit numbers in the finite general linear group

Jeffrey Chen, Jesse Selover

TL;DR

This work investigates positivity properties and modular polynomiality for $q$-rook and $q$-hit numbers associated with boards in the finite General Linear Group. By deploying inclusion-exclusion and an orbit-counting method, the authors show that residues of these counting functions in low degrees are polynomial in the variable $q-1$, and they establish a nonnegative linear coefficient for the $q$-hit number when expressed as $H_d(B,x+1)$ modulo $x^2$. They introduce generalized rook and hit numbers based on a small set of bi-colored graphs and derive explicit congruences for $M_d(B,q)$ and $H_d(B,q)$ modulo powers of $(q-1)$, including a complete polynomiality result modulo $(q-1)^6$. The paper further proves the nonnegativity of the linear coefficient in $H_d(B,x+1)$ for all boards and provides a framework for extending these results to higher residue classes, outlining both the opportunities and challenges ahead.

Abstract

We consider the problem of counting matrices over a finite field with fixed rank and support contained in a fixed set. The count of such matrices gives a $q$-analogue of the classical rook and hit numbers, known as the $q$-rook and $q$-hit numbers. They are known not to be polynomial in $q$ in general. We use inclusion-exclusion on the support of the matrices and the orbit counting method of Lewis et al. to show that the residues of these functions in low degrees are polynomial. We define a generalization of the classical rook and hit numbers which count placements of certain classes of graphs. These give us a formula for residues of the $q$-rook and $q$-hit numbers in low degrees. We analyze the residues of the $q$-hit number and show that the coefficient of $q-1$ in the $q$-hit number is always non-negative.

Positivity properties of $q$-hit numbers in the finite general linear group

TL;DR

This work investigates positivity properties and modular polynomiality for -rook and -hit numbers associated with boards in the finite General Linear Group. By deploying inclusion-exclusion and an orbit-counting method, the authors show that residues of these counting functions in low degrees are polynomial in the variable , and they establish a nonnegative linear coefficient for the -hit number when expressed as modulo . They introduce generalized rook and hit numbers based on a small set of bi-colored graphs and derive explicit congruences for and modulo powers of , including a complete polynomiality result modulo . The paper further proves the nonnegativity of the linear coefficient in for all boards and provides a framework for extending these results to higher residue classes, outlining both the opportunities and challenges ahead.

Abstract

We consider the problem of counting matrices over a finite field with fixed rank and support contained in a fixed set. The count of such matrices gives a -analogue of the classical rook and hit numbers, known as the -rook and -hit numbers. They are known not to be polynomial in in general. We use inclusion-exclusion on the support of the matrices and the orbit counting method of Lewis et al. to show that the residues of these functions in low degrees are polynomial. We define a generalization of the classical rook and hit numbers which count placements of certain classes of graphs. These give us a formula for residues of the -rook and -hit numbers in low degrees. We analyze the residues of the -hit number and show that the coefficient of in the -hit number is always non-negative.

Paper Structure

This paper contains 10 sections, 16 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background information
  3. Orbits of matrices
  4. Counting matrices by support
  5. Classifying orbits of size (q-1)(̂d+1) in m_d(B,q)
  6. Polynomiality of m_d(B,q) modulo (q-1)(̂d+6)
  7. Hit numbers modulo (q-1)2̂
  8. Finding H_d(B,x+1) modulo x^2
  9. Positivity in q-hit number coefficients
  10. Final Remarks

Key Result

Theorem 1

For a board $B \subseteq [n]\times[n]$, $H_d(B,x+1)$ is polynomial modulo $x^2$, and its coefficient of $x$ is non-negative.

Figures (2)

  • Figure 1: Three bi-colored graphs for $\mathop{\mathrm{\mathsf{ZG}}}\nolimits, \mathop{\mathrm{\mathsf{SG}}}\nolimits, \mathop{\mathrm{\mathsf{WRG}}}\nolimits,$ and $\mathop{\mathrm{\mathsf{WCG}}}\nolimits$
  • Figure 2: Boards used in the proof of Proposition \ref{['prop:orbitSize']}

Theorems & Definitions (44)

  • Theorem : Theorem \ref{['thm:q-coeff1-pos']}
  • Conjecture : Conjecture \ref{['conj:q-poly-all-coeffs']}
  • Proposition 2.1: lewis_rook_2020
  • Example 2.2
  • Remark 2.3
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • ...and 34 more