Parity biases in partitions and restricted partitions

Abstract

Let po(n) (resp. pe(n)) denote the number of partitions of n with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim and Lovejoy proved that po(n)>pe(n) for all n>2 and conjectured that do(n)>de(n) for all n>19 where do(n) (resp. de(n)) denote the number of partitions into distinct parts having more odd parts (resp. even parts) than even parts (resp. odd parts). In this paper we provide combinatorial proofs for both the result and the conjecture of Kim, Kim and Lovejoy. In addition, we show that if we restrict the smallest part of the partition to be 2, then the parity bias is reversed. That is, if qo(n) (resp. qe(n)) denote the number of partitions of n with more odd parts (resp. even parts) than even parts (resp. odd parts) where the smallest part is at least 2, then we have qo(n)7. We also look at some more parity biases in partitions with restricted parts.