author = {K. Banerjee},
title = {{Invariants of the quartic binary form and proofs of Chen's conjectures on inequalities for the partition function and the Andrews' spt function}},
language = {english},
abstract = {An extensive amount of study has been done on inequalities for the partition function, emerged primarily through works of Chen. In particular, the Tur'{a}n inequality and the higher order Tur'{a}n inequalities for $p(n)$ has been one of the most predominant theme. Among many others, one of the most notable one is Griffin, Ono, Rolen, and Zagier's result in which they proved that for every integer $d geq 1$, there exists an integer $N(d)$ such that the Jensen polynomial of degree $d$ and shift $n$ associated with the partition function, denoted by $J^{d,n}_p(x)$, has only distinct real roots for all $n geq N(d)$, earlier conjectured by Chen, Jia, and Wang and Ono independently. Later, Larson and Wagner have provided an estimate of upper bound for $N(d)$. This phenomena in turn implies that the discriminant of $J^{d,n}_p(x)$ is positive; i.e., $text{Disc}_{x}(J^{d,n}_p)>0$. For $d=2$, $text{Disc}_{x}(J^{2,n}_p)>0$ when $n geq N(2)=26$ is equivalent to the fact that $(p(n))_{n geq 26}$ is $log$-concave. In 2017, Chen undertook a comprehensive investigation on inequalities for $p(n)$ through the lens of invariant theory of binary forms of degree $n$. Positivity of the invariant of a quadratic binary form (resp. cubic binary form) associated with $p(n)$ reflects that the sequence $(p(n))_{n geq 26}$ satisfies the Tur'{a}n inequality (resp. $(p(n))_{n geq 95}$ satisfies the higher order Tur'{a}n inequality). Chen further studied on the two invariants for a quartic binary form where its coefficients are shifted values of integer partitions and conjectured four inequalities for $p(n)$. In this paper, we give explicit error bounds for the asymptotic expansion of the shifted partition function $p(n-ell)$ for any non-negative integer $ell$. As an application of these infinite family of inequalities, we confirm the conjectures of Chen. Moreover, three family of inequalities related to the partition function have been studied in this paper, namely, higher order Laguerre inequalities, higher order shifted differences, and higher order log-concavity. In context of higher order Laguerre inequalities for $p(n)$, we settle a conjecture of Wagner. For higher order shifted difference of $p(n)$, we extend a result of Gomez, Males, and Rolen. In context of higher order log-concavity for $p(n)$, we prove discuss on the asymptotic growth for the $r$-fold applications (with $rin {1,2,3}$) of the operator $mathcal{L}$ on $p(n)$ defined by $mathcal{L}(p(n))=p(n)^2-p(n-1)p(n+1)$ and propose a conjecture on infinite log-concavity in this regard. Furthermore, we will show how to construct a unified framework to prove partition function inequalities of the above types and discuss a few possible applications of such construction. Finally, we prove all the Chen's conjectures related to the inequalities for the Andrews' spt function, denoted by spt$(n)$, arising from invariants of quartic binary form using inequalities for the shifted partition function.},
year = {2023},
keywords = {the partition function, Andrews’ spt function, Hardy-Ramanujan-Rademacher for- mula, invariants of binary forms, combinatorial inequalities},
length = {55},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}