Races among products
  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. 
Alexander Berkovich and Keith Grizzell
math.CO math.NT