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

TL;DR

This paper investigates the properties of GBG-rank for t-cores, establishing bounds on its distinct values and expressing generating functions for 4-cores as eta-products, revealing new structural insights.

Contribution

It proves bounds on the number of GBG-rank values for t-cores and characterizes generating functions for 4-cores as eta-products, advancing understanding of core partitions.

Findings

Bound v(s,t) <= binomial(s+t,s)/(s+t) for coprime s,t

Equality holds when s is prime or t <= 2p_s

Generating functions for 4-cores are eta-products

Abstract

Let r_j(\pi,s) denote the number of cells, colored j, in the s-residue diagram of partition \pi. The GBG-rank of \pi mod s is defined as r_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 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<=2p_s, where p_s 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.