Distributions on partitions arising from Hilbert schemes and hook lengths

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 $$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$

Limitations to equidistribution in arithmetic progressions

“…Let h(a, b; n) count the number of hook lengths of length ℓ ≡ a (mod b) among all partitions of n. In this note we follow the techniques of [1] alongside Wright's Circle Method to determine our main theorem, showing that h(a, b; n) are asymptotically equidistributed as n → ∞, thereby finishing the story when taken alongside the results in [2].…”

“…In more recent times, Wright's Circle Method has been used in many papers in the literature the most prominent example that is relevant to the present paper is in [2] which was concerned with so-called t-hooks of partitions. Recall that a partition λ = (λ 1 , .…”

“…In [2], Bringmann, Craig, Ono, and the author studied the t-hooks of partitions (that is, hook lengths divisible by t ≥ 2) and their distribution properties over arithmetic progressions. However, the case t = 1 (i.e.…”

“…Here we recall a variant proved in [2] which is particularly easy-to-use and applicable to a wide range of generating functions.…”

A Note on the Equidistribution of 3-Colour Partitions

“…In recent years, one such problem has been the equidistribution properties of partition-theoretic objects over arithmetic progressions. For example, in [1] Bringmann, Craig, Ono, and the author studied the asymptotic equidistribution of Betti numbers of certain Hilbert schemes, as well as the non-equidistribution of t-hooks of partitions.…”