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Register a new userThe spt-Function of Andrews
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William Y.C. Chen
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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 $ngeq 95$ and for spt($n$) when $ngeq 108$.
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In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function $mathrm{spt}(n)$. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function $p(n)$ and $mathrm{spt}(n)$. Further, we strengthen one of the conjectures, and prove that for every $epsilon>0$ there is an effectively computable constant $N(epsilon) > 0$ such that for all $ngeq N(epsilon)$, we have begin{equation*} frac{sqrt{6}}{pi}sqrt{n},p(n)<mathrm{spt}(n)<left(frac{sqrt{6}}{pi}+epsilonright) sqrt{n},p(n). end{equation*} Due to the conditional convergence of the Rademacher-type formula for $mathrm{spt}(n)$, we must employ methods which are completely different from those used by Lehmer to give effective error bounds for $p(n)$. Instead, our approach relies on the fact that $p(n)$ and $mathrm{spt}(n)$ can be expressed as traces of singular moduli.
The Tur{a}n inequalities and the higher order Tur{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P{o}lya class. A real sequence ${a_{n}}$ is said to satisfy the Tur{a}n inequalities if for $ngeq 1$, $a_n^2-a_{n-1}a_{n+1}geq 0$. It is said to satisfy the higher order Tur{a}n inequalities if for $ngeq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2geq 0$. A sequence satisfying the Turan inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence ${ p(n)}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Tur{a}n inequalities for $ngeq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $ngeq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $mgeq 4$ there exists an integer $N(m)$ such that for $ngeq N(m) $, the polynomials $sum_{k=0}^m {mchoose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.
In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar, et al. but also unveils a new characteristic of such identities. As further applications, we establish a general form of Warnaars identity and a general $q$--series transformation associated with Bailey pairs via the use of the power series expansion of partial theta functions.
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In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis results on the rank and crank counts for ordinary partitions.
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