Number Of Overpartitions Research Articles

Refining overpartitions by properties of non-overlined parts

We study new classes of overpartitions of numbers based on the properties of non-overlined parts. Several combinatorial identities are established by means of generating functions and bijective proofs. We show that our enumeration function satisfies a pair of infinite Ramanujantype congruences modulo 3. Lastly, by conditioning on the overlined parts of overpartitions,we give a seemingly new identity between the number of overpartitions and a certain class of ordinary partition functions. A bijective proof for this theorem also includes a partial answer to a previous request for a bijection on partitions doubly restricted by divisibility and frequency.

Inequalities between overpartition ranks for all moduli

In this paper we give a full description of the inequalities that can occur between overpartition ranks modulo cge 2. If overline{N}(a,c,n) denotes the number of overpartitions of n with rank congruent to a modulo c, we prove that for any cge 7 and 0le a<ble leftlfloor frac{c}{2}rightrfloor we have overline{N}(a,c,n)>overline{N}(b,c,n) for n large enough. That the sign of the rank differences overline{N}(a,c,n)-overline{N}(b,c,n) depends on the residue class of n modulo c in the case of small moduli, such as c=6, is known due to the work of Ji et al. (J Number Theory 184:235–269, 2018) and Ciolan (Int J Number Theory 16(1):121–143, 2020). We show that the same behavior holds for cin {2,3, 4,5}.

On the number of l-regular overpartitions

An [Formula: see text]-regular overpartition of [Formula: see text] is an overpartition of [Formula: see text] into parts not divisible by [Formula: see text]. Let [Formula: see text] be the number of [Formula: see text]-regular overpartitions of [Formula: see text]. Andrews defined singular overpartitions counted by the partition function [Formula: see text]. It denotes the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts[Formula: see text] may be overlined. He proved that [Formula: see text] and [Formula: see text] are divisible by [Formula: see text]. In this paper, we aim to introduce a crank of [Formula: see text]-regular overpartitions for [Formula: see text] to investigate the partition function [Formula: see text]. We give combinatorial interpretations for some congruences of [Formula: see text] including infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text] as well as the congruences of Andrews for [Formula: see text] and [Formula: see text].

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 $\overline {C}_{6, 2}(n)$ .

Certain eta-quotients and $$\ell $$-regular overpartitions

Let $${\overline{A}}_{\ell }(n)$$ be the number of overpartitions of n into parts not divisible by $$\ell $$ . In this paper, we prove that $${\overline{A}}_{\ell }(n)$$ is almost always divisible by $$p_i^j$$ if $$p_i^{2a_i}\ge \ell $$ , where j is a fixed positive integer and $$\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}$$ with primes $$p_i>3$$ . We obtain a Ramanujan-type congruence for $${\overline{A}}_{7}$$ modulo 7. We also exhibit infinite families of congruences and multiplicative identities for $${\overline{A}}_{5}(n)$$ .

Explicit congruences modulo 2048 for overpartitions

Let $${\bar{p}}(n)$$ denote the number of overpartitions of n. Recently, a number of congruences modulo powers of 2, 3 and 5 have been discovered. The moduli for these congruences modulo powers of 2 discovered before ranged as high as 1024. In this paper, we establish explicit congruences modulo 2048 for $${\bar{p}}(n)$$ . In particular, we deduce some strange congruences modulo 2048 for $${\bar{p}}(n)$$ . For instance, we prove that for $$n\ge 0$$ , $$\begin{aligned} {\bar{p}}(1820n^2+1820n+455)\equiv 0\!\!\!\!\pmod {2048}. \end{aligned}$$

INFINITE FAMILIES OF CONGRUENCES FOR OVERPARTITIONS WITH RESTRICTED ODD DIFFERENCES

Let $\overline{t}(n)$ be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for $\overline{t}(n)$ modulo small powers of $2$ and $3$ have been established. We present two infinite families of congruences modulo $5$ and $27$ for $\overline{t}(n)$, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci. 129 (2019), Article 31].

Proof of a conjecture on a congruence modulo 243 for overpartitions

Let $$\bar{p}(n)$$ denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding $$\bar{p}(n)$$ . In particular, Xia discovered several infinite families of congruences modulo 9 and 27 for $$\bar{p}(n)$$ . Moreover, Xia conjectured that for $$n\ge 0$$ , $$\bar{p}(96n+76) \equiv 0\ (\mathrm{mod}\ 243)$$ . In this paper, we confirm this conjecture by using theta function identities and the (p, k)-parametrization of theta functions.

Some infinite families of congruences for overpartitions

Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. Recently, several infinite families of congruences modulo 12, 27 and 40 for [Formula: see text] have been established. In this paper, we investigate systematically some other infinite families of congruences modulo 12, 24 and 40 for [Formula: see text].

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 , , .

Andrews' singular overpartitions with odd parts

Recently singular overpartitions was defined and studied by G. E. Andrews. He showed that such partitions can be enumerated by $\overline{C}_{\delta,i}(n)$, the number of overpartitions of $n$ such that no part is divisible by $\delta$ and only parts $\equiv\pm i(mod \delta)$ may be overlined. In this paper, we establish several infinite families of congruences $\overline{CO}_{\delta,i}(n)$, the number of singular overpartitions of $n$ into odd parts such that no part is divisible by $\delta$ and only parts $\equiv\pm i\AFODMod{\delta}$ may be overlined. For example, for all $n\geq 0$ and $\alpha\geq0$, $\overline{CO}_{3,1}(4\cdot3^{\alpha+3}n+7\cdot3^{\alpha+2})\equiv 0 \pmod{8}$.

Congruences modulo 9 for singular overpartitions

In a recent work, Andrews introduced the new combinatorial objects called singular overpartitions. He proved that these singular overpartitions can be enumerated by the partition function [Formula: see text] which denotes the number of overpartitions of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. In this paper, we consider the function [Formula: see text] from an arithmetical point of view. We establish a number of Ramanujan-like congruences and a congruence relation modulo [Formula: see text] for [Formula: see text] by employing some generating function dissection techniques. With the aid of these congruences and the relation, we obtain two new infinite families of congruences modulo [Formula: see text] satisfied by [Formula: see text].

Congruences for $$\ell $$ ℓ -regular overpartitions and Andrews’ singular overpartitions

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and 9. Recently, Andrews introduced the partition function \(\overline{C}_{k, i}(n)\), called singular overpartition, 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 over-lined. He also proved that \(\overline{C}_{3, 1}(9n+3)\) and \(\overline{C}_{3, 1}(9n+6)\) are divisible by 3. In this paper, we prove that \(\overline{C}_{3, 1}(12n+11)\) is divisible by 144 which was conjectured to be true by Naika and Gireesh.

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].

Overpartition function modulo powers of 2

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, congruences modulo powers of 2 for \(\overline{p}(n)\) were widely studied. In this paper, we prove several new infinite families of congruences modulo powers of 2 for \(\overline{p}(n)\). For example, for \(\alpha \ge 1\) and \(n\ge 0\), $$\begin{aligned} \overline{p}(8\cdot 3^{4\alpha +4}n+5\cdot 3^{4\alpha +3})\equiv 0 \quad (\mathrm{mod}\,\,{2^8}). \end{aligned}$$

Overpartitions and ternary quadratic forms

Let \(\overline{p}(n)\) denote the number of overpartitions of n. Recently, Mahlburg showed that \(\overline{p}(n) \equiv 0 \pmod {64}\) and Kim showed that \(\overline{p}(n) \equiv 0 \pmod {128}\) for almost all integers n. In this paper, with the help of some ternary quadratic forms, we prove that \(\overline{p}(n) \equiv 0 \pmod {256}\) for almost all integers n, which was conjectured by Mahlburg.

Arithmetic properties of a partition pair function

For a positive integer n, let ped (n) be the number of partitions of n where the even parts are distinct, and [Formula: see text] be the number of overpartitions of n into odd parts. Moreover, let Q(n) denote the number of the partition pairs of n into two colors (say, red and blue), where the parts colored red satisfy restrictions of partitions counted by ped (n), while the parts colored blue satisfy restrictions of partitions counted by [Formula: see text]. We establish several congruences for Q(n). We also obtain an asymptotic formula for Q(n).