The metric space of limit laws for $q$-hook formulas

TL;DR

This paper explores the asymptotic behavior of coefficients in various q-hook formulas, revealing a rich space of limit laws including normal and non-normal distributions, and classifies their convergence properties.

Contribution

It extends previous work on q-hook formulas to new combinatorial objects, characterizes the space of all possible limit distributions, and introduces the DUSTPAN distribution class.

Findings

Coefficients are generally asymptotically normal.

Uncountably many non-normal limit laws exist.

Complete classification of limit distributions for the size statistic on plane partitions.

Abstract

In earlier work, Billey--Konvalinka--Swanson studied the asymptotic distribution of the coefficients of Stanley's $q$-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's $q$-hook-content formula related to semistandard tableaux and $q$-hook length formulas of Bj\"orner--Wachs related to linear extensions of labeled forests. We show that, while their coefficients are ``generically'' asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded $2$-norm. The closure of these distributions in the L\'evy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.