Parity biases in partitions and restricted partitions

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