Hook lengths in self-conjugate partitions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910322068029440 |
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| author | Amdeberhan, Tewodros Andrews, George E. Ono, Ken Singh, Ajit |
| author_facet | Amdeberhan, Tewodros Andrews, George E. Ono, Ken Singh, Ajit |
| contents | 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 $n_t(λ)$ is the number of size $t$ hooks in a partition $λ,$ then for even $t$ we have $$\sum_{λ\in \mathcal{SC}} x^{n_t(λ)} q^{\vertλ\vert} = (-q;q^2)_{\infty} \cdot ((1-x^2)q^{2t};q^{2t})_{\infty}^{\frac{t}2}. $$ As a consequence, if $a_t^*(n)$ is the number of such hooks among the self-conjugate partitions of $n,$ then for even $t$ we obtain the simple formula $$ a_t^*(n)=t\sum_{j\geq 1} q^*(n-2tj), $$ where $q^*(m)$ is the number of partitions of $m$ into distinct odd parts. As a corollary, we find that $t\mid a_t^*(n),$ which confirms a conjecture of Ballantine, Burson, Craig, Folsom, and Wen. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_02933 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hook lengths in self-conjugate partitions Amdeberhan, Tewodros Andrews, George E. Ono, Ken Singh, Ajit Combinatorics 05A15, 05A17, 11P81, 11P83 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 $n_t(λ)$ is the number of size $t$ hooks in a partition $λ,$ then for even $t$ we have $$\sum_{λ\in \mathcal{SC}} x^{n_t(λ)} q^{\vertλ\vert} = (-q;q^2)_{\infty} \cdot ((1-x^2)q^{2t};q^{2t})_{\infty}^{\frac{t}2}. $$ As a consequence, if $a_t^*(n)$ is the number of such hooks among the self-conjugate partitions of $n,$ then for even $t$ we obtain the simple formula $$ a_t^*(n)=t\sum_{j\geq 1} q^*(n-2tj), $$ where $q^*(m)$ is the number of partitions of $m$ into distinct odd parts. As a corollary, we find that $t\mid a_t^*(n),$ which confirms a conjecture of Ballantine, Burson, Craig, Folsom, and Wen. |
| title | Hook lengths in self-conjugate partitions |
| topic | Combinatorics 05A15, 05A17, 11P81, 11P83 |
| url | https://arxiv.org/abs/2312.02933 |