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We study the zero set of polynomials built from partition statistics, complementing earlier work in this direction by Boyer, Goh, Parry, and others. In particular, addressing a question of Males with two of the authors, we prove asymptotics for the values of $$t$$-hook polynomials away from an annulus and isolated zeros of a theta function. We also discuss some open problems and present data on other polynomial families, including those associated to deformations of Rogers-Ramanujan functions. 

Séminaire Lotharingien de Combinatoire - FPSAC 2023; Proceedings of the 35th International Conference on "Formal Power Series and Algebraic Combinatorics", July 17 - 21, 2023, University of California at Davis, USA; Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We 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 ≥ 2. To this end, we establish very general linear inequalities for the number of distinct partitions, which is also of independent interest. We also establish additional related partition bias results. 

Abstract Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on $${\mathbb {C}}^2$$ . For the Hilbert schemes, we prove that homology is equidistributed as $$n\to \infty $$ . For t -hooks, we prove distributions that are often not equidistributed. The cases where $$t\in \{2, 3\}$$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$