Parity bias in partitions

Let p(o)(n) denote the number of partitions of n with more odd parts than even parts and let p(e)(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 p(o)(n) - p(e)(n), which implies that p(o)(n) > p(e)(n) for all positive integers n not equal 2. Using combinatorial mappings we prove a stronger result, namely that for all n > 7 we have 2p(e)(n) < p(o)(n) < 3p(e)(n). Finally, using asymptotic methods we show that p(o)(n)/p(e)(n) -> 1 + root 2 as n -> infinity. We also examine related properties for two other types of partitions. (C) 2020 Elsevier Ltd. All rights reserved.