Singular Overpartitions Research Articles

On some congruences for unrestricted singular overpartitions

A specific case of partition called Andrews’ singular overpartitions, enumerated by [Formula: see text], is the number of overpartitions of [Formula: see text] where no summand is divisible by [Formula: see text] and the first occurrence of any summands which is [Formula: see text] can be marked. A number of authors have studied the properties of this partition. On this paper, we define a new partition which is a modification of singular overpartition, where you may chose any one of the summands which is [Formula: see text] to be marked. We enumerate that particular partition by [Formula: see text] and prove some congruences for [Formula: see text] on modulo [Formula: see text] and [Formula: see text], congruences for [Formula: see text] on modulo [Formula: see text], [Formula: see text] and [Formula: see text], and how likely it is for [Formula: see text] to be even.

Density results for the parity of (4k,k)-singular overpartitions

The (k,i)-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6k. In this paper we consider the parity of the number C‾k,i(n) of (k,i)-singular overpartitions of n. In particular, we give a sufficient condition on even values of k so that the values of C‾4k,k(n) are almost always even. Furthermore, we show that for odd values of k≤23, k≠19, certain subsequences of C‾4k,k(n) are almost always even.

Equalities for the 𝑟3-Crank of 3-Regular Overpartitions

Lovejoy introduced the partition function as the number of 𝑙-regular overpartitions of 𝑛. Andrews defined (𝑘, 𝑖)-singular overpartitions counted by the partition function , and pointed out that . Meanwhile, Andrews derived an interesting divisibility property that (mod 3). Recently, we constructed the partition statistic 𝑟𝑙-crank of 𝑙-regular overpartitions and give combinatorial interpretations for some congruences of as well as the congruences of Andrews. In this paper, we aim to prove some equalities for the 𝑟3-crank of 3-regular overpartitions.

Andrews Singular Overpartition Pairs with Odd Parts

Andrews Singular Overpartition Pairs with Odd Parts

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 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 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.

Singular Overpartitions and Partitions with Prescribed Hook Differences

Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (K, i)-singular overpartitions for integers K, i with $$ 1\le i<K/2$$. The definition of such singular overpartitions requires successive ranks, parity blocks and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in 1987. In this paper, employing hook differences, we generalize parity blocks. Using this combinatorial concept, we define $$(K,i,\alpha , \beta )$$-singular overpartitions for positive integers $$\alpha , \beta $$ with $$\alpha +\beta <K$$, and then we show some connections between such singular overpartitions and ordinary partitions.

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.

Divisibility of Andrews’ singular overpartitions by powers of 2 and 3

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 overlined. He also proved that $$\overline{C}_{3, 1}(9n+3)$$ and $$\overline{C}_{3, 1}(9n+6)$$ are divisible by 3 for $$n\ge 0$$ . Recently Aricheta proved that for an infinite family of k, $$\overline{C}_{3k, k}(n)$$ is almost always even. In this paper, we prove that for any positive integer k, $$\overline{C}_{3, 1}(n)$$ is almost always divisible by 2 $$^k$$ and 3 $$^k$$ .

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 Andrews singular overpartition pairs

Andrews defined the combinatorial objects called singular overpartitions denoted by [Formula: see text], which counts 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 investigate the arithmetic properties of Andrews singular overpartition pairs. Let [Formula: see text] be the number of overpartition pairs of [Formula: see text] in which no part is divisible by [Formula: see text] and only parts [Formula: see text] may be overlined. We will prove a number of Ramanujan like congruences and infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text], infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text], infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text].

Enumeration of partitions with prescribed successive rank parity blocks

Successive ranks of a partition, which were introduced by Atkin, are the difference of the lengths of the i-th row and the i-th column in the Ferrers graph. Recently, in the study of singular overpartitions, Andrews revisited successive ranks and parity blocks. Motivated by his work, we investigate partitions with prescribed successive rank parity blocks. The main result of this paper is the generating function of partitions with exactly d successive ranks and m parity blocks.

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 , , .

Some New Congruences for Andrews’ Singular Overpartition Pairs

In a recent work, Andrews defined the combinatorial objects called singular overpartitions denoted by $\overline {C}_{k,i}(n)$ , which count the number of overpartitions of n in which no part is divisible by k and only parts congruent to ± i modulo k may be overlined. Many authors have found congruences and infinite families of congruences modulo powers of 2 and 3. In this paper, we find some new infinite families of congruences for $\overline {C}^{6}_{1,2}(n)$ modulo 27 and congruences modulo 4 for $\overline {C}^{12}_{1,5}(n)$ , $\overline {C}^{9}_{3,3}(n)$ and $\overline {C}^{15}_{5,5}(n)$ .

New congruences for t-core partitions and Andrews' singular overpartitions

A partition of n is called a t-core partition of n if none of its hook numbers are multiples of t. Let the number of t-core partitions of n be denoted by at(n). Recently, G. E. Andrews defined combinatorial objects which he called (k,i) singular overpartitions, overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. Let the number of (k,i) singular overpartitions of n be denoted by C‾k,i(n). The object of this paper is to obtain new congruences modulo 2 for a15(n) and a23(n). We also obtain congruences modulo 2 for C‾92,23 and C‾60,15.

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 $\alpha\geq0$, $\overline{CO}_{3,1}(4\cdot3^{\alpha+3}n+7\cdot3^{\alpha+2})\equiv 0 \pmod{8}$.

Andrew's singular overpartitions without multiples of k

In a recent work G. E. Andrews gave definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters δ and i, can be enumerated by function Cδ,i(n), the number of overpartitions of n such that no part divisible by δ and only parts ≡ ∓ i (mod δ) may be overlined. In this paper, we establish several infinite families of congruences Ckδ,i(n), the number of singular overpartitions of n without multiples of k such that no part divisible by δ and only parts ≡ ∓ i (mod δ) may be overlined. For example, for all n ≥ 0 and α ≥ 0, C54,1(22α+5n + (7 ∙ 2α+3+1) ∕ 3) ≡ 0 (mod 22).

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].