On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations
| dc.creator | Berkovich, Alexander | |
| dc.creator | Garvan, Frank G. | |
| dc.date | 2004-02-26 | |
| dc.date.accessioned | 2026-07-07T05:05:46Z | |
| dc.date.available | 2026-07-07T05:05:46Z | |
| dc.description | In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0; 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enumerated by p[i](5n + 4) (i = 0, 2) into five equinumerous classes. In this paper we discuss three such statistics: the St-crank, the 2-quotient-rank and the 5-core-crank. The first one, while new, is intimately related to the Andrews-Garvan crank. The second one is in terms of the 2-quotient of a partition. The third one was introduced by Garvan, Kim and Stanton. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod 5. This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5. Finally, we discuss some new formulas for partitions that are 5-cores and discuss an intriguing relation between 3-cores and the Andrews-Garvan crank. | |
| dc.description | 24 pages, 2figures. This paper is greatly expanded version of an earlier paper which appeared as arXiv:math.CO/0401012 | |
| dc.identifier | https://arxiv.org/abs/math/0402439 | |
| dc.identifier | http://arxiv.org/abs/math/0402439 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70292 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | Primary 11P81, 11P83; Secondary 05A17, 05A19 | |
| dc.title | On the Andrews-Stanley Refinement of Ramanujan's Partition Congruence Modulo 5 and Generalizations | |
| dc.type | text |