[Abstract] Alexander Berkovich and Frank G. Garvan The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear.


Abstract: Let pi denote a partition into parts lambda1>lambda2>lambda3... In [BG] we defined BG-rank (pi) as

BG-rank(pi) = SUM(-1)j+1(1-(-1)lambdaj)/2.
This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let pj(n) denote a number of partitions of n with BG-rank=j. Here, we provide a combinatorial proof that
pj(5n+4)= 0 (mod 5), j in  Z,
by showing that the residue of the 5-core crank mod 5 (see [BG], [GKS]) divides the partitions enumerated by pj(5n+4) into five equal classes. This proof uses the orbit construction in [BG] and a new identity for the BG-rank. Let at,j(n) denote the number t-cores of n with BG-rank=j. In addition, we find eta-quotient representations for
SUM at,[(t+1)/4](n)qn   and  
SUM at,-[(t-1)/4](n)qn
when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a5,j(n), j=0,± 1.

The url of this page is http://www.math.ufl.edu/~frank/abstracts/bgrank.html.
Created by F.G. Garvan (fgarvan@ufl.edu) on Friday, March 03, 2006.
Last update made Sat Apr 28 09:39:04 EDT 2007.

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