The GBG-Rank and t-Cores I. Counting and 4-Cores
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 r0+r1*ws+r2*ws2+...+r(s-1)*ws(s-1), where ws=exp(2**I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2ps, where ps is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all eta-products.
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