A combinatorial correspondence between finite Euclidean geometries and symmetric subsets of $\mathbb{Z}/n\mathbb{Z}$

TL;DR

This paper establishes a combinatorial link between finite Euclidean geometries and symmetric subsets of cyclic groups, introduces dot-binomial coefficients, and explores their polynomial properties and relations to q-binomial coefficients.

Contribution

It introduces a novel combinatorial correspondence and expresses dot-binomial coefficients in terms of q-binomial coefficients and polynomials, expanding understanding of finite Euclidean geometries.

Findings

Dot-binomial coefficients can be expressed using q-binomial coefficients.

Dot-binomial coefficients are polynomials in q.

The paper reveals properties of the polynomials derived from dot-binomial coefficients.

Abstract

$q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients which give the number of $k$-dimensional subspaces in $\mathbb{F}_{q}^{n}$. When $q$ goes to $1$, this reverts to the binomial coefficients which measure the number of $k$-sets in $\left [ n \right ]$. Dot-analogues of $q$-binomial coefficients were studied by Yoo (2019) in order to investigate combinatorics of quadratic spaces over finite fields. The number of $k$-dimensional quadratic spaces of $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ which are isometrically isomorphic to $(\mathbb{F}_{q}^{k},x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2})$ can be also described as analogous to binomial coefficients, called the dot-binomial coefficients, $\binom{n}{k}_{d}$. In this paper, we study a combinatorial correspondence between this finite Euclidean geometries and symmetric subsets of $\mathbb{Z}/n\mathbb{Z}$. In addition, we show that dot-binomial coefficients can be expressed in terms of $q$-binomial coefficients and polynomials, and we prove that dot-binomial coefficients are polynomials in $q$. Furthermore, we study the properties of the polynomials given by the dot binomial coefficients $\binom{n}{k}_{d}$.