Races among products

TL;DR

This paper revisits a 1987 problem, providing an elementary injective proof that one partition count is always greater than or equal to another for all positive parameters, advancing understanding of partition inequalities.

Contribution

It offers a new elementary injective proof of a partition inequality related to a 1987 question, expanding the mathematical understanding of partition counts.

Findings

Proved P_1(L,y,n) >= P_2(L,y,n) for all L,n>0 and odd y>1

Established inequalities between specific partition functions

Provided an elementary proof approach for a classical problem

Abstract

We will revisit a 1987 question of Rabbi Ehrenpreis. Among many things, we will provide an elementary injective proof that P_1(L,y,n)>=P_2(L,y,n) for any L,n>0 and any odd y>1 . Here, P_1(L,y,n) denotes the number of partitions of n into parts congruent to 1, y+2, or 2y mod 2(y+1) with the largest part not exceeding 2(y+1)L-2 and P_2(L,y,n) denotes the number of partitions of n into parts congruent to 2, y, or 2y+1 mod 2(y+1) with the largest part not exceeding 2(y+1)L-1.