Binary partitions revisited

Abstract

The restricted binary partition function bk(n) enumerates the num-ber of ways to represent n as n = 2 a0 + 2 a1 + · · · + 2 aj with 0 ≤ a0 ≤ a1 ≤... ≤ aj < k. We study the question of how large a power of 2 divides the difference bk(2 r+2 n) − bk−2(2 r n) for fixed k ≥ 3, r ≥ 1, and all n ≥ 1. 1