The Gaussian polynomials, or \(q\)-binomial coefficients, are defined by \begin{equation}\label{gnreq1} g(n,r;q):=\left [ \begin{matrix} n\\ r \end{matrix} \right ] := \left [ \begin{matrix} n\\ r \end{matrix} \right ]_q := \begin{cases} \displaystyle{ \frac{(q;q)_n}{(q;q)_r (q;q)_{n-r}}}, &0\leq r \leq n,\\ &\\ 0, & \text{ otherwise.} \end{cases} \end{equation} One connection with integer partitions is contained in the following theorem.

Theorem. For non-negative integers \(M\) and \(N\), \begin{equation}\label{15partsgpolyseq} g(M+N,M;q)=\left [ \begin{matrix} M+N\\ M \end{matrix} \right ]=\sum_{n\geq 0}p(N,M,n)q^n, \end{equation} where \(p(N,M,n)\) denotes the number of partitions of the integer \(n\) into at most \(M\) parts, each \(\leq N\).

This implies the following corollary.

Corollary For non-negative integers \(M\) and \(N\), \begin{equation}\label{15partsgpolyseq2} \sum_{n\geq 0}p(N,M,n) =\left ( \begin{matrix} M+N\\ M \end{matrix} \right ), \end{equation} where the right side is the usual binomial coefficient.

The identity in the above theorem may be used to give combinatorial proofs of the two special cases of the $q$-binomial theorem.

Corollary For each non-negative integer \(N\), \begin{equation}\label{c141} \sum_{m=0}^{N} \left[ \begin{matrix} N\\ m \end{matrix} \right] z^mq^{m(m+1)/2}=(-zq;q)_{N}, \end{equation} \begin{equation}\label{c142} \sum_{m=0}^{\infty}q^m \left[ \begin{matrix} N+m-1\\ m \end{matrix} \right]z^m=\frac{1}{(zq;q)_{N}},\,\,|z|< 1,\,\,|q| < 1. \end{equation}
A proof may be found in [1], Section 7.3.

[1] Andrews, G. E.; Eriksson, K. (2004), Integer partitions. Cambridge University Press, Cambridge, x+141 pp

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