The spt-Function of Andrews

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 $n\geq 108$.