Races among products

In this paper we revisit a 1987 question of Rabbi Ehrenpreis. Among many things, we provide an elementary injective proof thatP1(L,y,n)⩾P2(L,y,n) for any L,n>0 and any odd y>1. Here, P1(L,y,n) denotes the number of partitions of n into parts congruent to 1, y+2, or 2y(mod2y+2) with the largest part not exceeding (2y+2)L−2 and P2(L,y,n) denotes the number of partitions of n into parts congruent to 2, y, or 2y+1(mod2y+2) with the largest part not exceeding (2y+2)L−1.