Infinite Families Of Congruences Research Articles

Ramanujan congruences for fractional partition functions

For rational α \alpha , the fractional partition functions p α ( n ) p_\alpha (n) are given by the coefficients of the generating function ( q ; q ) ∞ α (q; q)_\infty ^\alpha . When α = − 1 \alpha = -1 , one obtains the usual partition function. Congruences of the form p ( ℓ n + c ) ≡ 0 ( mod ℓ ) p(\ell n + c)\equiv 0 \pmod {\ell } for a prime ℓ \ell and integer c c were studied by Ramanujan. Such congruences exist only for ℓ ∈ { 5 , 7 , 11 } \ell \in \{5, 7, 11\} . Chan and Wang recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as p 57 61 ( 17 2 n − 3 ) ≡ 0 ( mod 17 2 ) p_\frac {57}{61}(17^2 n - 3) \equiv 0 \pmod {17^2} .

Parity of the coefficients of certain eta-quotients

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of m-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all m≤28, we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible.All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area.

On some infinite families of congruences for [j, k]-partitions into even parts distinct

On some infinite families of congruences for [j, k]-partitions into even parts distinct

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].

Arithmetic properties of 3-regular partitions with distinct odd parts

Let $$pod_3(n)$$ denote the number of 3-regular partitions of n with distinct odd parts (and even parts are unrestricted). In this article, we prove an infinite family of congruences for $$pod_3(n)$$ using the theory of Hecke eigenforms. We also study the divisibility properties of $$pod_3(n)$$ using arithmetic properties of modular forms.

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 $${\overline{A}}_{5}(n)$$ .

Newman's identities, lucas sequences and congruences for certain partition functions

AbstractLet r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] and for k ≥ 0, \[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] where spt(n) denotes Andrews's smallest parts function.

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.

Infinite Families of Congruences for 3-Regular Partitions with Distinct Odd Parts

Let $$pod_3(n)$$ denote the number of 3-regular partitions with distinct odd parts (and even parts are unrestricted) of a non-negative integer n. In this paper, we present infinite families of Ramanujan-type congruences modulo 2 and 3 for $$pod_3(n)$$ .

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 the number of tagged parts over partitions with designated summands

Recently, Lin introduced a new partition function [Formula: see text], which counts the total number of tagged parts over all partitions of [Formula: see text] with designated summands. Lin also proved some congruences modulo [Formula: see text] and [Formula: see text] for [Formula: see text]. In this paper, we shall present two new infinite families of congruences modulo [Formula: see text] 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 , , .

Ramanujan-type congruences for ℓ-regular partitions modulo 3, 5, 11 and 13

Let [Formula: see text] be the number of [Formula: see text]-regular partitions of [Formula: see text]. Recently, Hou et al. established several infinite families of congruences for [Formula: see text] modulo [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the vanishing property given by Hou et al., we prove an infinite family of congruence for [Formula: see text] modulo [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text], we obtain three infinite families of congruences for [Formula: see text] modulo [Formula: see text] and [Formula: see text] respectively using the theory of Hecke eigenforms.

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 powers of 5 for two restricted bipartitions

Let \(B_{5}(n)\) denote the number of 5-regular bipartitions of n. We establish some Ramanujan-type congruences like \(B_{5}(4n+3) \equiv 0\) (mod 5) and many infinite families of congruences for \(B_{5}(n)\) modulo higher powers of 5 such as $$\begin{aligned} B_{5}\left( 5^{2k-1}n+\frac{2\cdot 5^{2k-1}-1}{3}\right) \equiv 0 \pmod {5^k}. \end{aligned}$$ We also apply the same method to obtain some similar results for another type of bipartition function. Meanwhile, we give a new interesting interlinked q-series identity related with Rogers–Ramanujan continued fraction, which answers a question of M. Hirschhorn.

Congruences modulo 27 for cubic partition pairs

Let b(n) denote the number of cubic partition pairs of n. This paper aims to study the congruences for b(n) modulo 27. We first establish three Ramanujan type congruences. Then many infinite families of congruences are presented. Finally, we propose two conjectures on the congruences for b(n) modulo 49,81 and 243.

Extensive parity analysis of broken 5-diamond partitions

In this paper, we investigate congruences for [Formula: see text] modulo 2 and characterize the parity of [Formula: see text] and [Formula: see text] according to the arithmetic property of [Formula: see text]. As a consequence, we obtain various Ramanujan type congruences for [Formula: see text]. We also extend these results to several infinite families of congruences.

Arithmetic properties of partition triples with odd parts distinct

Let pod -3(n) denote the number of partition triples of n where the odd parts in each partition are distinct. We find many arithmetic properties of pod -3(n) involving the following infinite family of congruences: for any integers α ≥ 1 and n ≥ 0, [Formula: see text] We also establish some arithmetic relations between pod (n) and pod -3(n), as well as some congruences for pod -3(n) modulo 7 and 11.

ARITHMETIC PROPERTIES OF PARTITION QUADRUPLES WITH ODD PARTS DISTINCT

Let $\text{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\text{pod}_{-4}(n)$ including the following infinite family of congruences: for any integers ${\it\alpha}\geq 1$ and $n\geq 0$, $$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$ We also establish some internal congruences and some Ramanujan-type congruences modulo 2, 5 and 8 satisfied by $\text{pod}_{-4}(n)$.

Arithmetic Properties of a Restricted Bipartition Function

A bipartition of $n$ is an ordered pair of partitions $(\lambda,\mu)$ such that the sum of all of the parts equals $n$. In this article, we concentrate on the function $c_5(n)$, which counts the number of bipartitions $(\lambda,\mu)$ of $n$ subject to the restriction that each part of $\mu$ is divisible by $5$. We explicitly establish four Ramanujan type congruences and several infinite families of congruences for $c_5(n)$ modulo $3$.