\[\sum_{\sigma\in\mathfrak{S}_{n}}q^{\mathop{\mathrm{inv}}(\sigma)}=\sum_{\sigma%
\in\mathfrak{S}_{n}}q^{\mathop{\mathrm{maj}}(\sigma)}=\prod_{j=1}^{n}\frac{1-q%
^{j}}{1-q}.\]