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01.On Even Integers Not of the Form 2k + σ(m) (pp. 263-268)
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The GBC Rank and t-Cores I. Counting and 4-Cores (pp. 237-252) $45.00
Authors:  Alexander Berkovich and Frank G. Garvan
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. 

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The GBC Rank and t-Cores I. Counting and 4-Cores (pp. 237-252)