Hook length biases and general linear partition inequalities

Abstract: Connections between representation theory and the theory of integer partitions are well-known, and hook lengths of partitions sometimes play important roles in this context. Partly motivated by these connections, we consider the total number of hooks of fixed length in odd versus distinct partitions. We establish bias results, and show that there are more hooks of length 2, respectively 3, in all odd partitions of $n$ than in all distinct partitions of $n$, and make the analogous conjecture for arbitrary hook length $t \geq 2$. We also establish and conjecture further related hook biases, asymptotics, and congruences in our work. An integral component to our proof is a linear inequality involving $q(n)$, the number of distinct parts partitions of $n$. We also establish effective linear inequalities for $q(n)$ in great generality, a result which is of independent interest. Our methods are analytic (i.e., Wright’s circle method, modularity, $q$-series transformations, asymptotic methods) and combinatorial in nature, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, and $q$-series. This is joint work by Cristina Ballantine (College of the Holy Cross), Hannah Burson (University of Minnesota), William Craig (University of Cologne), Amanda Folsom (Amherst College), and Boya Wen (University of Wisconsin).

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