The BG-rank of a partition and its applications
| dc.creator | Berkovich, Alexander | |
| dc.creator | Garvan, Frank G. | |
| dc.date | 2006-02-16 | |
| dc.date | 2007-04-28 | |
| dc.date.accessioned | 2026-07-07T07:58:27Z | |
| dc.date.available | 2026-07-07T07:58:27Z | |
| dc.description | Let πbe a partition. In [2] we defined BG-rank(π) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a number of partitions (t-cores) of n with BG-rank=j. Here, we provide an elegant combinatorial proof that 5|p_j(5n+4) by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by p_j(5n+4) into five equal classes. This proof uses the orbit construction in [2] and new identity for BG-rank. In addition, we find eta-quotient representation for the generating functions for coefficients a_{t,floor((t+1)/4)}(n), a_{t,-floor((t-1)/4)}(n) when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a_{5,j}(n) with j=0,1,-1. | |
| dc.description | 20 pages. This version has an expanded section 7, where we defined gbg-rank and stated a number of appealing results. We added a new reference. This paper will appear in Adv. Appl. Math | |
| dc.identifier | https://arxiv.org/abs/math/0602362 | |
| dc.identifier | http://arxiv.org/abs/math/0602362 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128016 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | 11P81, 11P83, 05A17, 05A19 | |
| dc.title | The BG-rank of a partition and its applications | |
| dc.type | text |