Asymptotic formulas for stacks and unimodal sequences

TL;DR

This paper surveys known results and introduces new asymptotic formulas for various unimodal sequences, stacks, and related combinatorial structures, connecting them to partitions and solving open problems in statistical mechanics.

Contribution

It provides new asymptotic formulas for unimodal sequences and stacks, filling gaps in existing literature and addressing an open problem in statistical mechanics.

Findings

Derived new asymptotic formulas for unimodal sequences and stacks.

Connected combinatorial structures like partitions, Frobenius symbols, and stacks.

Solved an open problem in statistical mechanics related to asymptotics.

Abstract

We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions. These results describe combinatorial properties, generating functions, and asymptotic formulas for the enumeration functions. We also prove several new asymptotic results that fill in the notable missing cases from the literature, including an open problem in statistical mechanics due to Temperley. Furthermore, we explain the combinatorial and asymptotic relationship between partitions, Andrews' Frobenius symbols, and stacks with summits.