Hook lengths in self-conjugate partitions

Abstract

In 2010, G.-N. Han obtained the generating function for the number of size t t hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even t t . If n t ( λ ) n_t(\lambda ) is the number of size t t hooks in a partition λ \lambda and S C \mathcal {SC} denotes the set of self-conjugate partitions, then for even t t we have ∑ λ ∈ S C x n t ( λ ) q | λ | = ( − q ; q 2 ) ∞ ⋅ ( ( 1 − x 2 ) q 2 t ; q 2 t ) ∞ t 2 . \begin{equation*} \sum _{\lambda \in \mathcal {SC}} x^{n_t(\lambda )} q^{\vert \lambda \vert } = (-q;q^2)_{\infty } \cdot ((1-x^2)q^{2t};q^{2t})_{\infty }^{\frac {t}{2}}. \end{equation*} As a consequence, if a t ⋆ ( n ) a_t^{\star }(n) is the number of such hooks among the self-conjugate partitions of n n , then for even t t we obtain the simple formula a t ⋆ ( n ) = t ∑ j ≥ 1 q ⋆ ( n − 2 t j ) , \begin{equation*} a_t^{\star }(n)=t\sum _{j\geq 1} q^{\star }(n-2tj), \end{equation*} where q ⋆ ( m ) q^{\star }(m) is the number of partitions of m m into distinct odd parts. As a corollary, we find that t ∣ a t ⋆ ( n ) t\mid a_t^{\star }(n) , which confirms a conjecture of Ballantine, Burson, Craig, Folsom and Wen.