Let \(S_n\) be the symmetric group of order \(n\) and \(\# \sigma\) be the sign of the permutation \(\sigma\in S_n\). When \(\sigma=(p_0\quad p_1\cdots p_{n-1})\), define \(I(\sigma)\) by \[ I(\sigma) :=\{l\in\mathbb Z: l_j\equiv j-p_j\pmod n, j=1,\cdots, n-1\}. \] Set \[ F_n:=F_n(y_1,\cdots, y_{n-1},q):= \sum_{\sigma\in S_n} (-1)^{\# \sigma}\sum_{l\in I(\sigma)} y_1^{l_1}y_2^{l_2}\cdots y_{n-1}^{l_{n-1}}q^{A(\mathbf{l})}, \] where \[ A(\mathbf{l})= A(l_1,\cdots,l_{n-1}) =\frac{1}{2}\left(\sum_{r=1}^{n-1} l_r^2+\left(\sum_{r=1}^{n-1} l_r\right)^2\right), \] and \[ \begin{aligned} G_n:= G_n(y_1,\cdots, y_{n-1},q)&:=\prod_{m=1}^\infty\{\prod_{1\leq j\leq n-1} [(1-y_jq^{mn-n+j})(1-y_j^{-1}q^{mn-j})]\\ &\times \prod_{1\leq j