Multiplication theorems for self-conjugate partitions. (English) Zbl 1498.05024
Summary: In 2011, G.-N. Han and K. Q. Ji [Trans. Am. Math. Soc. 363, No. 2, 1041–1060 (2011; Zbl 1227.05040)] proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by A. Berkovich and F. G. Garvan [Trans. Am. Math. Soc. 358, No. 2, 703–726 (2006; Zbl 1088.11073)]. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A17 | Combinatorial aspects of partitions of integers |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05E05 | Symmetric functions and generalizations |
| 05E10 | Combinatorial aspects of representation theory |
| 11P81 | Elementary theory of partitions |
Keywords:
hook-length formulas; BGP-ranks; integers partitions; Littlewood decomposition; core partitionsReferences:
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