Overpartition Function Research Articles

Certain eta-quotients and arithmetic density of Andrews' singular overpartitions

In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function C‾k,i(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. Andrews also proved two beautiful Ramanujan type congruences modulo 3 satisfied by C‾3,1(n). Later on, Aricheta proved that for an infinite family of ℓ, C‾3ℓ,ℓ(n) is almost always divisible by 2. In this article, for an infinite subfamily of ℓ considered by Aricheta, we prove that C‾3ℓ,ℓ(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾3ℓ,ℓ(n) is almost always divisible by arbitrary powers of 3 when ℓ=3,6,12,24. Proofs of our density results rely on the modularity of certain eta-quotients which arise naturally as generating functions for the Andrews' singular overpartition functions.

K-Regular partitions and overpartitions with bounded part differences

Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions with bounded part differences by using Dousse and Kim’s results on (q, z)-overGaussian polynomials.

Inequalities for the dth residual crank moments of overpartitions

In 2009, Bringmann, Lovejoy and Osburn defined two analogues of the crank function for overpartitions, namely the first residual crank and second residual crank. For a positive integer [Formula: see text], we introduce the notion of a [Formula: see text]th residual crank function for overpartitions, examine the positive moments of these generalized crank functions while varying [Formula: see text], and prove some inequalities.

Inequalities for the overpartition function

Let $$\overline{p}(n)$$ denote the overpartition function. Engel showed that for $$n\ge 2$$ , $$\overline{p}(n)$$ satisfy the Turan inequalities, that is, $$\overline{p}(n)^2-\overline{p}(n-1)\overline{p}(n+1)>0$$ for $$n\ge 2$$ . In this paper, we prove several inequalities for $$\overline{p}(n)$$ . Moreover, motivated by the work of Chen, Jia and Wang, we find that the higher order Turan inequalities of $$\overline{p}(n)$$ can also be determined.

SUMS OF SQUARES AND PARTITION CONGRUENCES

AbstractFor positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

Finding Modular Functions for Ramanujan-Type Identities

This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions \( \mathit(\overline{p}) \)(5n + 2) and \( \mathit(\overline{p}) \)(5n + 3) and Andrews–Paule’s broken 2-diamond partition functions \( \triangle_{2} \)(25n+14) and \( \triangle_{2} \)(25n+24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews’ singular overpartition functions \( \mathit(\overline{Q}_{3,1}) \)(9n + 3) and \( \mathit(\overline{Q}_{3,1}) \)(9n + 6) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.

A Simple Proof of a Congruence for a Series Involving the Little q-Jacobi Polynomials

We give a simple and a more explicit proof of a mod 4 congruence for a series involving the little q-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

A Truncated Theta Identity of Gauss and Overpartitions into Odd Parts

We examine two truncated series derived from a classical theta identity of Gauss. As a consequence, we obtain two infinite families of inequalities for the overpartition function $$\overline{p_o}(n)$$ counting the number of overpartitions into odd parts. We provide partition-theoretic interpretations of these results.

Congruences for the coefficients of the powers of the Euler Product

Let $$p_k(n)$$ be given by the series expansion of the k-th power of the Euler Product $$\prod _{n=1}^{\infty }(1-q^n)^k=\sum _{n=0}^{\infty }p_k(n)q^{n}$$. By investigating the properties of the modular equations of the second and the third order under the Atkin U-operator, we determine the generating functions of $$p_{8k}(2^{2\alpha } n +\frac{k(2^{2\alpha }-1)}{3})$$$$(1\le k\le 3)$$ and $$p_{3k}(3^{2\beta }n+\frac{k(3^{2\beta }-1)}{8})$$$$(1\le k\le 8)$$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo m, we obtain infinite families of congruences for $$p_k(n)$$ modulo any $$m\ge 2$$, where $$1\le k\le 24$$ and 3|k or 8|k. Based on these congruences for $$p_k(n)$$, infinite families of congruences for many partition functions such as the overpartition function, t-core partition functions and $$\ell $$-regular partition functions are easily obtained.

Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions

Abstract Singular overpartition functions were defined by Andrews. Let Ck,i(n) denote the number of (k, i)-singular overpartitions of n, which counts the number of overpartitions of n in which no part is divisible by k and only parts ±i (mod k) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for Ck,i(n) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs (k, i). In this paper, we prove some congruences modulo powers of 2 for C48, 6(n) and C48, 18(n).

Congruences for two restricted overpartitions

Congruences for partitions have received a great deal of attention in literature. Recently, Bringmann et al. (Electron J. Combin. 22(3) (2015) Paper 3.17, 16 pp.) studied overpartitions with restricted odd differences. In this paper, we present a number of Ramanujan-type congruences for these restricted overpartition functions.

An overpartition analogue of q-binomial coefficients, II: Combinatorial proofs and (q,t)-log concavity

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an m×n rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization [m+nn]‾q,t of Gaussian polynomials, which is also a (q,t)-analogue of Delannoy numbers. First we obtain finite versions of classical q-series identities such as the q-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite theta function and prove the (q,t)-log concavity of [m+nn]‾q,t. We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of [m+nn]‾q,t.

New infinite families of congruences for Andrews’ (K, I)-singular overpartitions

In a recent work, Andrews defined the singular overpartition functions, denoted by , which count the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. Moreover, many congruences modulo 3, 9 and congruences modulo powers of 2 for were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pair (k, i). In this paper, we proved new infinite families of congruences modulo 27 for and infinite families of congruences modulo 4 and 8 for , , .

An overpartition analogue of partitions with bounded differences between largest and smallest parts

We study the generating function for overpartitions with bounded differences between largest and smallest parts, which is analogous to a result of Breuer and Kronholm on integer partitions. We also connect this problem with over q-binomial coefficients introduced by Dousse and Kim.

Finite difference of the overpartition function

Let p(n) denote the integer partition function. Good conjectured that Δrp(n) alternates in sign up to a certain value n=n(r), and then it stays positive. Gupta showed that for any given r and sufficiently large n, Δrp(n)>0. Odlyzko proved this conjecture and gave an asymptotic formula for n(r). Then, Almkvist, Knessel and Keller gave many contributions for the exact value of n(r). For the finite difference of log⁡p(n), DeSalvo and Pak proved that 0≤−△2log⁡p(n−1)≤log⁡(1+1n) and conjectured a sharper upper bound for −△2log⁡p(n). Chen, Wang and Xie proved this conjecture and showed the positivity of (−1)r−1△rlog⁡p(n), and further gave an upper bound for (−1)r−1△rlog⁡p(n). As for the overpartition function p‾(n), Engel recently proved that p‾(n) is log-concave for n≥2, that is, −△2log⁡p‾(n)≥0 for n≥2. Motivated by these results, in this paper we will prove the positivity of finite differences of the overpartition function and give an upper bound for △rp‾(n). Then we show that for any given r≥1, there exists a positive number n(r) such that (−1)r−1△rlog⁡p‾(n)>0 for n>n(r), where △ is the difference operator with respect to n. Moreover, we give an upper bound for (−1)r−1△rlog⁡p‾(n).

Partitions into a small number of part sizes

We study [Formula: see text], the number of partitions of [Formula: see text] into [Formula: see text] part sizes, and find numerous arithmetic progressions where [Formula: see text] and [Formula: see text] take on values divisible by 2 and 4. Expanding an earlier work, we show [Formula: see text] for ([Formula: see text]) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function [Formula: see text] in the first three progressions (the fourth is known), and thereby show that [Formula: see text] in each of these progressions as well, and discuss the relationship between these congruences in more generality. We end with open questions in this area.

Overpartition function modulo 16 and some binary quadratic forms

Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. In this paper, we will give a complete determination of [Formula: see text] modulo [Formula: see text] by relating it to some binary quadratic forms; this will generalize the result of Kim on [Formula: see text] modulo [Formula: see text]. Moreover, we give infinitely many new Ramanujan-like congruences for overpartition function modulo [Formula: see text].

Log-concavity of the overpartition function

We prove that the overpartition function $$ \overline{p}(n)$$ is log-concave for all $$ n\ge 2 $$ . The proof is based on Sills-Rademacher-type series for $$ \overline{p}(n)$$ and inspired by DeSalvo and Pak’s proof for the partition function.

Overpartitions with Restricted Odd Differences

We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.

DEVELOPMENT OF LOVEJOY AND OSBURN’SOVERPARTITION FUNCTIONS

In 2008, Lovejoy and Osburn defined the generating function for .In 2009, Byungchan Kim defined the generating function for .This paper shows how to discuss the generating functions for and . Byungchan Kim also defined with increasing relation and overpartitioncongruences mod 4,8 and 64. In 2006, Berndt found the relation has two values with certain restrictions and various formulae by the common term .This paper shows how to prove the four Theorems about overpartitions modulo 8.These Theorems satisfy the arithmetic properties of the overpartition function modulo 8.