The GBC Rank and t-Cores I. Counting and 4-Cores (pp. 237-252)
Authors: Alexander Berkovich and Frank G. Garvan
Abstract: Let rj (π, s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of π mod s is defined as
GBG-rank(π, s) = s−1 Σ j=0 rj(π, s)e2πi/s j
We prove that for (s, t) = 1
v(s, t) ≤ (s+t/s)/s+t , where v(s, t) denotes the number of distinct values that the GBG-rank of a t-core mod s may assume. The above inequality becomes an equality when s is prime or when s is composite and t ≤ 2ps, where ps is the smallest prime divisor of s. We show that the generating functions for 4-cores with prescribed GBG-rank mod 3 value are all eta-quotients.