Consider the next essence $$ B_N (r, q) =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \frac{\Gamma [N + \tau_i - i +1]}{\Gamma [N + \tau_i - i +1+q]} $$ where $d(\tau)$ is dimension of representation of a permutation group $S_r$ and $\tau =(\tau_1,...,\tau_{l(\tau)})$ is a partition of $\tau \vdash r$ and $r\leq N$ .
For example, here are the first few coefficients: \begin{equation} B_N(0,q) \ = \ B_N(r,0) \ = \ 1 \ , \label{QSUN_r0} \end{equation} \begin{equation} B_N(1,q) \ = \ \frac{1}{\Delta_1} \ , \label{QSUN_r1} \end{equation} \begin{equation} B_N(2,q) \ = \ \frac{1}{\Delta_2}[N^2 y-1] \ , \label{QSUN_r2} \end{equation} \begin{equation} B_N(3,q) \ = \ \frac{1}{\Delta_3}[N^4 y^2 -N^2(3 y+2)+4] \ , \label{QSUN_r3} \end{equation} and \begin{eqnarray} &&\Delta_r \ = \ y^{p_0} (N^2 y^2-1)^{p_1} ...(N^2 y^2 - (r-1)^2)^{p_{r-1}} \ , \end{eqnarray} where \begin{equation} p_m^r \equiv p_m = \left[ \frac{\sqrt{4 r + m^2} - m}{2} \right], \ \ \ y = 1+ q/N \end{equation}
Also it is known result for $q=1$ \begin{equation} B_N(r,1) \ = \ \frac{r! N!}{(r+N)!} \ L_r^{(N)}(1) \ , \label{BSUN_q1} \end{equation} where $L_r^{(N)}(x)$ is the generalized Laguerre polynomial.
Early, I found general representation of this object in the limit $N \to \infty $.
I`m interesting in explicit formula or any connection to known function or objects.
More simple case here: New identity for sum over Young diagram of symmetric group? , SYT and contents of a partition
