Races among products

In this paper we revisit a 1987 question of Rabbi Ehrenpreis. Among many things, we 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 2 y ( mod 2 y + 2 ) with the largest part not exceeding ( 2 y + 2 ) L − 2 and P 2 ( L , y , n ) denotes the number of partitions of n into parts congruent to 2, y, or 2 y + 1 ( mod 2 y + 2 ) with the largest part not exceeding ( 2 y + 2 ) L − 1 .