Parity bias in partitions

Abstract

Let po(n) denote the number of partitions of n with more odd parts than even parts and let pe(n) denote the number of partitions of n with more even parts than odd parts. Using q-series transformations we find a generating function for po(n)−pe(n), which implies that po(n)>pe(n) for all positive integers n≠2. Using combinatorial mappings we prove a stronger result, namely that for all n>7 we have 2pe(n)<po(n)<3pe(n). Finally, using asymptotic methods we show that po(n)∕pe(n)→1+2 as n→∞. We also examine related properties for two other types of partitions. © 2020 Elsevier Ltd