Parity biases in partitions and restricted partitions

TL;DR

This paper proves combinatorially that certain parity biases in partitions hold universally and explores how these biases change under restrictions like minimal part size, revealing nuanced behaviors in partition structures.

Contribution

It provides combinatorial proofs for parity bias results in partitions and investigates how restrictions alter these biases, including reversing the inequalities.

Findings

Proved that $p_o(n)>p_e(n)$ for all $n>2$.

Confirmed that $d_o(n)>d_e(n)$ for all $n>19$.

Showed that restricting the smallest part to 2 reverses the bias for $n>7$.

Abstract

Let $p_{o}(n)$ (resp. $p_{e}(n)$) 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 $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured that $d_{o}(n)>d_{e}(n)$ for all $n>19$ where $d_{o}(n)$ (resp. $d_{e}(n)$) 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 $q_{o}(n)$ (resp. $q_{e}(n)$) 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 $q_o(n)<q_e(n)$ for all $n>7$. We also look at some more parity biases in partitions with restricted parts.