The BG-rank of a partition and its applications

Abstract

Let π denote a partition into parts λ 1 ⩾ λ 2 ⩾ λ 3 ⩾ ⋯ . In a 2006 paper we defined BG-rank ( π ) as BG-rank ( π ) = ∑ j ⩾ 1 ( − 1 ) j + 1 1 − ( − 1 ) λ j 2 . This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p j ( n ) denote the number of partitions of n with BG-rank = j . Here, we provide a combinatorial proof that p j ( 5 n + 4 ) ≡ 0 ( mod 5 ) , j ∈ Z , by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by p j ( 5 n + 4 ) into five equal classes. This proof uses the orbit construction from our previous paper and a new identity for the BG-rank. Let a t , j ( n ) denote the number of t-cores of n with BG-rank = j . We find eta-quotient representations for ∑ n ⩾ 0 a t , ⌊ t + 1 4 ⌋ ( n ) q n and ∑ n ⩾ 0 a t , − ⌊ t − 1 4 ⌋ ( n ) q n , when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a 5 , j ( n ) , j = 0 , ± 1 .