On the hook length biases of the $2$- and $3$-regular partitions
Wenxia Qu, Wenston J. T. Zang
TL;DR
This work investigates hook length biases in $2$- and $3$-regular partitions by studying $b_{t,i}(n)$, the total number of $i$-hooks across all $t$-regular partitions of $n$. It combines generating-function techniques with combinatorial interpretations and explicit bijections to compare hook-length counts, establishing a positive bias $b_{3,2}(n)-b_{3,1}(n)\ge0$ for large $n$ and revealing a nuanced behavior for the monotone bias conjecture in the even/odd $k$-regimes. The authors prove the conjecture holds for $k=4$ and $k=6$ but show that for odd $k\ge3$ the corresponding inequality fails for infinitely many $n$, while also confirming the conjecture only for certain even $k$ and proposing a refined statement for the even case. Overall, the paper clarifies how hook-length biases behave across $t$-regular partitions and contributes to the broader understanding of partition inequalities via $q$-series methods and bijective combinatorics.
Abstract
Let $b_{t,i}(n)$ denote the total number of the $i$ hooks in the $t$-regular partitions of $n$. Singh and Barman (J. Number Theory { 264} (2024), 41--58) raised two conjectures on $b_{t,i}(n)$. The first conjecture is on the positivity of $b_{3,2}(n)-b_{3,1}(n)$ for $n\ge 28$. The second conjecture states that when $k\ge 3$, $b_{2,k}(n)\ge b_{2,k+1}(n)$ for all $n$ except for $n= k+1$. In this paper, we confirm the first conjecture. {Moreover, we show that for any odd $k\ge 3$, the second conjecture fails for infinitely many $n$.} {Furthermore, we verify that the second conjecture holds for $k=4$ and $6$.} We also propose a conjecture on the even case $k$, which is a modification of Singh and Barman's second conjecture.