On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo \(5\) and generalizations. (English) Zbl 1088.11073
Let \(p(n)\) be the number of unrestricted partitions of the positive integer \(n\). Let \(O(\pi)\) be the number of odd parts of the partition \(\pi\) and let \(\pi'\) be the conjugate of \(\pi\). The Stanley rank of a partition \(\pi\) is defined as \(srank(\pi)=O(\pi)=O(\pi')\) and let \(p_i(n)\), \(i=0,2\) be the number of partitions of \(n\) with \(srank\equiv i\bmod 4\). In this paper, it is provided a combinatorial interpretation (in terms of partition statistics) of the equality \(p_0(5n+4)\equiv p_2(5n+4)\equiv 0\bmod 5\) that was proved by Andrews. The authors also obtained Andrews’ result as a corollary of a new refinement of a Ramanujan’s result stating that \(p(5n+4)\equiv 0\bmod 5\) (this stronger refinement uses a new partition statistic).
Reviewer: J. L. Ramirez Alfonsin (Paris)
MSC:
| 11P81 | Elementary theory of partitions |
| 11P83 | Partitions; congruences and congruential restrictions |
| 05A17 | Combinatorial aspects of partitions of integers |
| 05A19 | Combinatorial identities, bijective combinatorics |
Keywords:
ranks Stanley’s statisticSoftware:
StanBij.mwsReferences:
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