Overpartition Function Research Articles

Explicit families of congruences for the overpartition function

In this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo m for m=3,5,7,11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m = 3, 5, 7, 11$$\\end{document}, and finite families for m=13,17,19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m = 13, 17, 19$$\\end{document}.

A further look at the overpartition function modulo $$2^4$$ and $$2^5$$

A further look at the overpartition function modulo $$2^4$$ and $$2^5$$

Congruence Properties of a Class of Constrained Overpartition Functions

Congruence Properties of a Class of Constrained Overpartition Functions

Inequalities for the overpartition function arising from determinants

Let p‾(n) denote the overpartition function. This paper presents the 2-log-concavity property of p‾(n) by considering a more general inequality of the following form|p‾(n)p‾(n+1)p‾(n+2)p‾(n−1)p‾(n)p‾(n+1)p‾(n−2)p‾(n−1)p‾(n)|>0, which holds for all n≥42.

Efficient computation of the overpartition function and applications

In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences, whose existence is predicted by Treneer and a few of which were first discovered by Ryan, Scherr, Sirolli and Treneer.

Polynomization of the Liu–Zhang inequality for the overpartition function

Polynomization of the Liu–Zhang inequality for the overpartition function

Higher order log-concavity of the overpartition function and its consequences

AbstractLet ${\overline{p}}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher-order log-concavity property of the overpartition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove log-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient ${\overline{p}}(n-1){\overline{p}}(n+1)/{\overline{p}}(n)^2$ up to a certain order. This enables us to additionally prove 2-log-concavity and higher Turán inequalities with a unified approach.

Positivity of the determinants of the partition function and the overpartition function

In this paper, we give an iterated approach to concern with the positivity of det ( p ( n − i + j ) ) 1 ≤ i , j ≤ k , \begin{equation*} \det \ (p(n-i+j))_{1\leq i,j\leq k}, \end{equation*} where p ( n ) p(n) is the partition function. We first apply a general method to prove that for given k 1 , k 2 , m 1 , m 2 k_1,k_2,m_1,m_2 , one can find a threshold N ( k 1 , k 2 , m 1 , m 2 ) N(k_1,k_2,m_1,m_2) such that for n > N ( k 1 , k 2 , m 1 , m 2 ) n>N(k_1,k_2,m_1,m_2) , | p ( n − k 1 + m 1 ) a m p ; p ( n + m 1 ) a m p ; p ( n + m 1 + m 2 ) p ( n − k 1 ) a m p ; p ( n ) a m p ; p ( n + m 2 ) p ( n − k 1 − k 2 ) a m p ; p ( n − k 2 ) a m p ; p ( n − k 2 + m 2 ) | > 0. \begin{equation*} \begin {vmatrix} p(n-k_1+m_1) & p(n+m_1) & p(n+m_1+m_2)\\ p(n-k_1) & p(n) & p(n+m_2)\\ p(n-k_1-k_2) & p(n-k_2) & p(n-k_2+m_2) \end{vmatrix}>0. \end{equation*} Based on this result, we will prove that for n ≥ 656 n\geq 656 , det ( p ( n − i + j ) ) 1 ≤ i , j ≤ 4 > 0 \det \ (p(n-i+j))_{1\leq i,j\leq 4}>0 . Employing the same technique, we will show that determinants ( p ¯ ( n − i + j ) ) 1 ≤ i , j ≤ k ({\bar p}(n-i+j))_{1\leq i,j\leq k} are positive for k = 3 and 4 k=3 \text { and } 4 for overpartition p ¯ ( n ) {\bar p}(n) . Furthermore, we will give an outline of how to prove the positivity of det ( p ( n − i + j ) ) 1 ≤ i , j ≤ k \det \ (p(n-i+j))_{1\leq i,j\leq k} for general k k .

New Combinatorial Interpretations for the Partitions into Odd Parts Greater than One

In this paper, we consider $Q_1(n)$ to be the number of partitions of $n$ into odd parts greater than one and provide new combinatorial interpretations for $Q_1(n)$. New linear relations involving Euler's partition function $p(n)$ and the overpartition function $\overline{p}(n)$ are obtained in this context.

Inequalities for higher order differences of the logarithm of the overpartition function and a problem of Wang–Xie–Zhang

Let {overline{p}}(n) denote the overpartition function. In this paper, our primary goal is to study the asymptotic behavior of the finite differences of the logarithm of the overpartition function, i.e., (-1)^{r-1}Delta ^r log {overline{p}}(n), by studying the inequality of the following form log(1+C(r)nr-1/2-1+C1(r)nr)<(-1)r-1Δrlogp¯(n)<log(1+C(r)nr-1/2)forn≥N(r),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\log \\Bigl (1+\\dfrac{C(r)}{n^{r-1/2}}-\\dfrac{1+C_1(r)}{n^{r}}\\Bigr ){} & {} <(-1)^{r-1}\\Delta ^r \\log {\\overline{p}}(n) \\\\ {}{} & {} <\\log \\Bigl (1+\\dfrac{C(r)}{n^{r-1/2}}\\Bigr )\\ \ ext {for}\\ n \\ge N(r), \\end{aligned}$$\\end{document}where C(r), C_1(r), text {and} N(r) are computable constants depending on the positive integer r, determined explicitly. This solves a problem posed by Wang, Xie and Zhang in the context of searching for a better lower bound of (-1)^{r-1}Delta ^r log {overline{p}}(n) than 0. By settling the problem, we are able to show that limn→∞(-1)r-1Δrlogp¯(n)=π2(12)r-1n12-r.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lim _{n\\rightarrow \\infty }(-1)^{r-1}\\Delta ^r \\log {\\overline{p}}(n) =\\dfrac{\\pi }{2}\\Bigl (\\dfrac{1}{2}\\Bigr )_{r-1}n^{\\frac{1}{2}-r}. \\end{aligned}$$\\end{document}

On the Ramanujan-type congruences modulo 8 for the overpartitions into odd parts

Let denote the number of overpartitions of n into odd parts. In this paper, we provide a complete characterization of Ramanujan-type congruences modulo 8 for the overpartition function considering the number of odd positive divisors of and the relations of the form with and

Log-convexity and the overpartition function

Let overline{p}(n) denote the overpartition function. In this paper, we obtain an inequality for the sequence Delta ^{2}log root n-1 of {overline{p}(n-1)/(n-1)^{alpha }} which states that log(1+3π4n5/2-11+5αn11/4)<Δ2logp¯(n-1)/(n-1)αn-1<log(1+3π4n5/2)forn≥N(α),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}&\\log \\biggl (1+\\frac{3\\pi }{4n^{5/2}}-\\frac{11+5\\alpha }{n^{11/4}}\\biggr )< \\Delta ^{2} \\log \\ \\root n-1 \\of {\\overline{p}(n-1)/(n-1)^{\\alpha }}\\\\&< \\log \\biggl (1+\\frac{3\\pi }{4n^{5/2}}\\biggr ) \\ \\ \ ext {for}\\ n \\ge N(\\alpha ), \\end{aligned}$$\\end{document}where alpha is a non-negative real number, N(alpha ) is a positive integer depending on alpha , and Delta is the difference operator with respect to n. This inequality consequently implies log -convexity of bigl {root n of {overline{p}(n)/n}bigr }_{n ge 19} and bigl {root n of {overline{p}(n)}bigr }_{n ge 4}. Moreover, it also establishes the asymptotic growth of Delta ^{2} log root n-1 of {overline{p}(n-1)/(n-1)^{alpha }} by showing underset{n rightarrow infty }{lim } Delta ^{2} log root n of {overline{p}(n)/n^{alpha }} = dfrac{3 pi }{4 n^{5/2}}.

Laguerre inequalities for discrete sequences

The Turán inequality, the Laguerre inequality and their m-rd generalizations have been proved to be closely relative with the Laguerre-Pólya class and Riemann hypothesis. Since these two inequalities are equivalent to log-concavity of the discrete sequences, we consider whether their generalizations hold for discrete sequences. Recently, Chen, Jia and Wang proved that the partition function satisfies the Turán inequality of order 2 and thus the 3-rd Jensen polynomials associated with the partition function have only real zeros. Griffin, Ono, Rolen and Zagier proved the n-th Jensen polynomials associated with the Maclaurin coefficients of the function in the Laguerre-Pólya class and the partition function have only real zeros except finite terms. In this paper, we show the Laguerre inequality of order 2 is true for the partition function, the overpartition function, the Bernoulli numbers, the derangement numbers, the Motzkin numbers, the Fine numbers, the Franel numbers and the Domb numbers.

Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

Kim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to a pmod {c} for cne 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

New truncated theorems for three classical theta function identities

In 2012, Andrews and Merca derived a truncated version of Euler’s pentagonal number theorem. Their work inspired several mathematicians to work on truncated theta series including Guo and Zeng, who examined two other classical theta series identities of Gauss. In this paper, revisiting these three theta series identities of Euler and Gauss, we obtain new truncated theorems. As corollaries of our results, we obtain infinite families of linear inequalities involving the partition function, the overpartition function and the pod function. These inequalities yield the positivity result of Andrews and Merca on the partition function as well as a conjecture on the overpartition function, which was posed by Andrews–Merca and Guo–Zeng, and proved independently by Mao and Yee. We will also provide a unified combinatorial treatment for our results.

Overpartition analogues for the generalized Rogers–Ramanujan identities of Bressoud

In the past two decades, there have been a lot of research centred around overpartitions, some of which concern overpartition analogues of Rogers–Ramanujan type identities. In this paper, we present Rogers–Ramanujan type overpartition identities by considering Bressoud’s even moduli generalization of the Rogers–Ramanujan identity and its overpartition analogue of Chen, Sang and Shi given in 2015. We first introduce another overpartition function C¯k,a(n) and show that C¯k,a(n) equals the overpartition function B¯k,a,0(n) of Chen, Sang and Shi. Next, we study parity constrains on parts of overpartitions. Recently, Sang, Shi and Yee obtained Rogers–Ramanujan type identities for overpartitions by adding some parity constraints to even or odd parts of overpartitions Chen, Sang and Shi introduced in 2013. We make some modifications and add constraints to even or odd parts of overpartitions counted by C¯k,a(n) and B¯k,a,0(n) obtaining further Rogers–Ramanujan type overpartition identities.

Generalizations of the Andrews–Yee identities associated with the mock theta functions $$\omega (q)$$ and $$\nu (q)$$

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third-order mock theta functions $$\omega (q)$$ and $$\nu (q)$$ , thereby extending their earlier results with the second author. Generalizing the Andrews–Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang’s three-variable reciprocity theorem. A relation between a new restricted overpartition function $$p^{*}(n)$$ and a weighted partition function $$p_*(n)$$ is obtained from one of the special cases of our second theorem.

New density results and congruences for Andrews' singular overpartitions

Andrews introduced the singular overpartition function C‾k,i(n) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. In this article, we study the divisibility properties of C‾4k,k(n) and C‾6k,k(n) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C‾4k,k(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾6k,k(n) is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by C‾4⋅2α,2α(n) and C‾4⋅3⋅2α,3⋅2α(n).

On a nonlinear relation for computing the overpartition function

Abstract In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p (n). Computing p (n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p (n) that requires only the values of p (k) with k ≤ n/2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p (n) to obtain a simple and fast computation of the value of p (n). This new method uses only (large) integer arithmetic and it is simpler to program.

DIVISIBILITY OF CERTAIN SINGULAR OVERPARTITIONS BY POWERS OF AND

AbstractAndrews introduced the partition function $\overline {C}_{k, i}(n)$ , called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$ . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$ .