Hook length biases in t-regular and t-core partitions

Abstract: Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen, hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Recently, Singh and Barman opened the door to hook length biases in $t$-regular partitions as well.

The objective of this talk is two fold. First, we present a previously unobserved connection of hook-lengths in $t$-regular partitions with certain distinct parts partitions. Second, we extend the theory of hook length biases to $t$-core partitions. For example, let $a_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-core partitions of $n$, then we find that $a_{3,1}(n) \ge a_{3,2}(n) \ge a_{3,4}(n)$ and $a_{4,1}(n) \ge a_{4,3}(n)$ for all $n$. Most of the methods employed in this work are combinatorial. Joint work with Talukdar; and Baruah, Das, and Mahanta.

number theory

Audience: researchers in the discipline

( paper )

Combinatorial and additive number theory seminar (CANT 2026)