  $q$-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 $\binom{n}{k}_{d}$.
