The BG-rank of a partition and its applications

2006/02/16 by Berkovich, Alexander, Garvan, Frank G.
#05A17 #05A19 #11P81 #11P83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0602362

Abstract

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.

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