Given an integer partition $\lambda$ and its Young diagram $Y_{\lambda}$, let $h_{\lambda}(i,j)$ stand for the corresponding hook length of the cell $(i,j)\in Y_{\lambda}$. Write $\lambda\vdash n$ for $\lambda$ a partition of $n$.

Recall the Gaussian binomials denoted by $$\mathbf{\binom{n}k_q}.$$

QUESTION.If $[q^k]F(q)=$ the coefficient of $q^k$ in the polynomial $F(q)$, is this true? $$\sum_{\lambda\vdash n}q^{h_{\lambda}(1,1)} =\sum_{i,j=0}^nq^j\cdot[q^{n-j}]\mathbf{\binom{j-1}i_q}.$$

**POSTSCRIPT.** If $p(n)$ is the number of partitions of $n$ then clearly we have
$$p(n)=\sum_{j=1}^n\sum_{i=0}^{j-1}\,\,\, [q^{n-j}]\mathbf{\binom{j-1}i_q}.$$