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Let r_j(\pi,s) denote the number of cells, colored j, in the s-residuediagram of partition \pi. The GBG-rank of \pi mod s is defined asr_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*\Pi*I/s). We willprove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes anumber of distinct values that GBG-rank mod s of t-core may assume. The aboveinequality becomes an equality when s is prime or when s is composite andt<=2p_s, where p_s is a smallest prime divisor of s. We will show that thegenerating functions for 4-cores with the prescribed values of GBG-rank mod 3are all eta-products.

The GBG-Rank and t-Cores I. Counting and 4-Cores