The spt-Function of Andrews

TL;DR

This survey reviews recent advances in the study of the spt-function, including congruences, proofs of conjectures, asymptotic formulas, and conjectures on inequalities, highlighting its connections to partition theory and invariant theory.

Contribution

It summarizes recent developments, including proofs of conjectures, generalizations, asymptotic results, and new conjectures on inequalities related to the spt-function.

Findings

Congruence properties of spt(n) established by multiple researchers.

A constructive proof of the Andrews-Dyson-Rhoades conjecture.

Asymptotic formulas for spt(n) derived by Ahlgren, Andersen, and Rhoades.

Abstract

The spt-function spt($n$) was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. In this survey, we summarize recent developments in the study of spt($n$), including congruence properties established by Andrews, Bringmann, Folsom, Garvan, Lovejoy and Ono et al., a constructive proof of the Andrews-Dyson-Rhoades conjecture given by Chen, Ji and Zang, generalizations and variations of the spt-function. We also give an overview of asymptotic formulas of spt($n$) obtained by Ahlgren, Andersen and Rhoades et al. We conclude with some conjectures on inequalities on spt($n$), which are reminiscent of those on $p(n)$ due to DeSalvo and Pak, and Bessenrodt and Ono. Furthermore, we observe that, beyond the log-concavity, $p(n)$ and spt($n$) satisfy higher order inequalities based on polynomials arising in the invariant theory of binary forms. In particular, we conjecture that the higher order Tur\'{a}n inequality $4(a_n^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_na_{n+1}-a_{n-1}a_{n+2})^2>0$ holds for $p(n)$ when $n\geq 95$ and for spt($n$) when $n\geq 108$.