# Reading off top hook-lengths in partitions

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 $$p(n)=\sum_{j=1}^n\sum_{i=0}^{j-1}\,\,\, [q^{n-j}]\mathbf{\binom{j-1}i_q}.$$

• Looking at the generating function at findstat.org/St000459 and then clicking on "search the OEIS" yields that this was observed in the OEIS (oeis.org/A049597) in 2008, but no reference to a proof is given. – Christian Stump May 7 '19 at 8:56

Let $$j$$ be the size of the $$(1,1)$$-hook, and $$i+1$$ the number of rows of this hook. To complete this hook to a partition of $$n$$, we must place to its southeast a partition of $$n-j$$ with at most $$j-i-1$$ columns and at most $$i$$ rows. The number of such partitions is $$[q^{n-j}]\boldsymbol{{j-1\choose i}}$$, and the proof follows.