| Title: |
Hook lengths in self-conjugate partitions. |
| Authors: |
Amdeberhan, Tewodros1 (AUTHOR), Andrews, George E.2 (AUTHOR), Ono, Ken3 (AUTHOR), Singh, Ajit3 (AUTHOR) |
| Source: |
Proceedings of the American Mathematical Society, Series B. 7/2/2024, Vol. 11, p345-357. 13p. |
| Subject Terms: |
*PARTITION functions, *GENERATING functions, *HOOKS, *PARTITIONS (Mathematics) |
| Abstract: |
In 2010, G.-N. Han obtained the generating function for the number of size t hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even t. If n_t(\lambda) is the number of size t hooks in a partition \lambda and \mathcal {SC} denotes the set of self-conjugate partitions, then for even t we have \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^{\star }(n) is the number of such hooks among the self-conjugate partitions of n, then for even t we obtain the simple formula \begin{equation*} a_t^{\star }(n)=t\sum _{j\geq 1} q^{\star }(n-2tj), \end{equation*} where q^{\star }(m) is the number of partitions of m into distinct odd parts. As a corollary, we find that t\mid a_t^{\star }(n), which confirms a conjecture of Ballantine, Burson, Craig, Folsom and Wen. [ABSTRACT FROM AUTHOR] |
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