SU\G(𝔸)/K·U
Sign Up for free Log In

ON $5^k$ -REGULAR PARTITIONS MODULO POWERS OF $5$

Authors

B. Hemanthkumar, D. S. Gireesh
Type: Article
Publication Date: 2025-04-14
Citations: 0
DOI: https://doi.org/10.1017/s0004972725000139

Abstract

Abstract In this work, we investigate the arithmetic properties of $b_{5^k}(n)$ , which counts the partitions of n where no part is divisible by $5^k$ . By constructing generating functions for $b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.

Topics

Locations

  • Bulletin of the Australian Mathematical Society
    View

Arithmetic properties of $5$-regular partitions into distinct parts

2024-11-05
Nayandeep Deka Baruah , Abhishek Sarma
A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into … A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prime_5(n)$. We provide full characterization of the parity of $b^\prime_5(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b^\prime_5(5n+1))$ is lacunary modulo any arbitrary positive powers of 5.
View PDF Chat

Arithmetic Properties of a Restricted Bipartition Function

2015-07-01
Jian Liu , Andrew Y. Z. Wang
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 … 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 $3$.
View PDF Chat

Distribution of the k-regular partition function modulo composite integers M

2022-01-01
Yiwen Lu , Xuejun Guo
Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, … Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton function $b_k(n)$ and integer $M$ coprime to $6$, there are infinitely many Ramanujan-type congruences of $b_k(n)$ modulo $M$.

Congruences modulo powers of 5 for $k$-colored partitions

2017-01-01
Dazhao Tang
Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite … Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for $p_{-k}(n)$ with $k=2, 6$, and $7$. For example, for all integers $n\geq0$ and $\alpha\geq1$, we prove that \begin{align*} p_{-2}\left(5^{2\alpha-1}n+\dfrac{7\times5^{2\alpha-1}+1}{12}\right) &\equiv0\pmod{5^{\alpha}} \end{align*} and \begin{align*} p_{-2}\left(5^{2\alpha}n+\dfrac{11\times5^{2\alpha}+1}{12}\right) &\equiv0\pmod{5^{\alpha+1}}. \end{align*}
View PDF Chat

Divisibility and distribution of 5-regular partitions

2022-01-01
Qi-Yang Zheng
In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the theory of modular forms, we prove several theorems on the divisibility and distribution properties of $b_5(n)$ modulo … In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the theory of modular forms, we prove several theorems on the divisibility and distribution properties of $b_5(n)$ modulo prime $m\geq5$. In particular, we prove that there are infinitely many Ramanujan-type congruences modulo prime $m\geq5$.

The Localization Method Applied to $k$-Elongated Plane Partitions and Divisibility by 5

2022-01-01
Koustav Banerjee , Nicolas Allen Smoot
The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. We undertake a comprehensive study of congruence families for $d_k(n)$ … The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. We undertake a comprehensive study of congruence families for $d_k(n)$ modulo powers of 5, with $n$ and $k$ in respective linear progressions. In particular, we give an infinite congruence family for $d_k(n)$ modulo powers of 5. This family cannot be proved by the classical methods, i.e., the techniques used to prove Ramanujan's congruence families for $p(n)$. Indeed, the proof employs the recently developed localization method, and utilizes a striking internal algebraic structure which has not yet been seen in the proof of any congruence family. We believe that this discovery poses important implications on future work in partition congruences.
View PDF Chat

Congruences for 5-regular partitions modulo powers of 5

2016-03-07
Liuquan Wang

New congruences for 4,6-regular partitions modulo primes

2023-01-01
Qi-Yang Zheng
The main result of the paper is the existence of an infinitely many families of Ramanujan-type congruences for $b_4(n)$ and $b_6(n)$ modulo primes $m \geq 2$ and $m \geq 5$, … The main result of the paper is the existence of an infinitely many families of Ramanujan-type congruences for $b_4(n)$ and $b_6(n)$ modulo primes $m \geq 2$ and $m \geq 5$, respectively. We provide new examples of congruences for $b_4(n)$ and $b_6(n)$. Moreover, we find two infinite explicit infinite families of congruences for $b_4(n)$ modulo $3$.
View PDF Chat

ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS

2009-07-02
Michael D. Hirschhorn , James A. Sellers
Abstract In a recent paper, Calkin et al . [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and … Abstract In a recent paper, Calkin et al . [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers 8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b 5 ( n ), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n .
View PDF Chat

On $k$-restricted overpartitions

2019-08-01
Uha Isnaini
We introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first … We introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first give the generating function and establish the $5$-dissections of $k$-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo $5$. Finally, we pose some problems for future work.

Congruences modulo 8 for $$(2,\, k)$$ ( 2 , k ) -regular overpartitions for odd $$k > 1$$ k > 1

2017-12-14
Chandrashekar Adiga , M. S. Mahadeva Naika , D. Ranganatha +1 more
In this paper, we study various arithmetic properties of the function $$\overline{p}_{2,\,\, k}(n)$$ , which denotes the number of $$(2,\,\, k)$$ -regular overpartitions of n with odd $$k > 1$$ … In this paper, we study various arithmetic properties of the function $$\overline{p}_{2,\,\, k}(n)$$ , which denotes the number of $$(2,\,\, k)$$ -regular overpartitions of n with odd $$k > 1$$ . We prove several infinite families of congruences modulo 8 for $$\overline{p}_{2,\,\, k}(n)$$ . For example, we find that for all non-negative integers $$\beta , n$$ and $$k\equiv 1\pmod {8}$$ , $$\overline{p}_{2,\,\, k}(2^{1+\beta }(16n+14))\equiv ~0\pmod {8}$$ .
View PDF Chat

The Andrews-Stanley partition function and $p(n)$: congruences

2010-12-20
Holly Swisher
R. Stanley formulated a partition function $t(n)$ which counts the number of partitions $\pi$ for which the number of odd parts of $\pi$ is congruent to the number of odd … R. Stanley formulated a partition function $t(n)$ which counts the number of partitions $\pi$ for which the number of odd parts of $\pi$ is congruent to the number of odd parts in the conjugate partition $\pi ’$ (mod 4). In light of G. E. Andrews’ work on this subject, it is natural to ask for relationships between $t(n)$ and the usual partition function $p(n)$. In particular, Andrews showed that the (pmod 5) Ramanujan congruence for $p(n)$ also holds for $t(n)$. In this paper we extend his observation by showing that there are infinitely many arithmetic progressions $An + B$ such that for all $n\geq 0$, \[ t(An+B) \equiv p(An+B) \equiv 0 \pmod {l^j} \] whenever $l\geq 5$ is prime and $j\geq 1$.
View PDF Chat

On 5-regular bipartitions with even parts distinct

2018-09-04
M. S. Mahadeva Naika , T. Harishkumar
View PDF Chat

On $(4, 5)$-regular bipartitions with odd parts distinct

2019-07-01
M. S. Mahadeva Naika , T. Harishkumar
In his work, K. Alladi considered the partition function $pod(n)$, the number of partitions of an integer $n$ with odd parts distinct (the even parts are unrestricted). He obtained a … In his work, K. Alladi considered the partition function $pod(n)$, the number of partitions of an integer $n$ with odd parts distinct (the even parts are unrestricted). He obtained a series expansion for the product generating function of these partitions. Later Hirschhorn and Sellers obtained some internal congruences involving the infinite families and Ramanujan's type congruences for $pod(n)$. Let $B_{4, 5}(n)$ denote the number of $(4, 5)$-regular bipartitions of a positive integer $n$ with odd parts distinct. In this paper, we establish many infinite families of congruences modulo powers of $2$ for $B_{4, 5}(n)$.

Congruences modulo powers of 5 for k-colored partitions

2017-11-21
Dazhao Tang
View PDF Chat

Congruences for $\ell$-regular partitions and bipartitions

2020-04-01
Su-Ping Cui , Nancy S. S. Gu
Define F ( q ) : = ∑ n = − ∞ ∞ ( − 1 ) δ n ( a n + b ) q ( c n 2 … Define F ( q ) : = ∑ n = − ∞ ∞ ( − 1 ) δ n ( a n + b ) q ( c n 2 + d n ) ∕ 2 , which includes Ramanujan’s theta function as a special case. We establish a dissection identity for this function, and use it to derive congruence properties for the coefficients of F ( q ) . As an application we deduce several infinite families of congruences for ℓ -regular partitions and ℓ -regular bipartitions. In addition, we give a new proof of Ramanujan’s congruence for the unrestricted partition function modulo 5 .

On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$

2022-01-01
George E. Andrews , James A. Sellers , Fares Soufan
In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $\phi_k(n)$ and $c\phi_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized … In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $\phi_k(n)$ and $c\phi_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of that Memoir, Andrews proved a number of Ramanujan--like congruences satisfied by specific functions within these two families. In the years that followed, numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ In this brief note, our goal is to identify an {\bf infinite} family of values of $k$ such that $\phi_k(n)$ is even for all $n$ in a specific arithmetic progression; in particular, our primary goal in this work is to prove that, for all positive integers $\ell,$ all primes $p\geq 5,$ and all values $r,$ $0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews' Memoir, classical $q$--series results, and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalized Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

Some Observations on Modulo 5 Congruences for 2-Color Partitions

2017-01-01
Suparno Ghoshal , Sourav Sen Gupta
The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one … The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one of the 2 colors appears only in parts that are multiples of $k$. In this paper, we record the complete characterization of the modulo 5 congruence relation $p_{k}(25n + 24 - k) \equiv 0 \pmod{5}$ for $k \in \{1, 2, \ldots, 24\}$, in connection with the 2-color partition function $p_k(n)$, providing references to existing results for $k \in \{1, 2, 3, 4, 7, 8, 17\}$, simple proofs for $k \in \{5, 10, 15, 20\}$ for the sake of completeness, and counter-examples in all the remaining cases. We also propose an alternative proof in the case of $k = 4$, without using the Rogers-Ramanujan ratio, thereby making the proof considerably simpler compared to the proof by Ahmed, Baruah and Ghosh Dastidar (JNT 2015).
View PDF Chat

Some Observations on Modulo 5 Congruences for 2-Color Partitions

2017-12-28
Suparno Ghoshal , Sourav Sen Gupta
The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one … The 2-color partitions may be considered as an extension of regular partitions of a natural number $n$, with $p_{k}(n)$ defined as the number of 2-colored partitions of $n$ where one of the 2 colors appears only in parts that are multiples of $k$. In this paper, we record the complete characterization of the modulo 5 congruence relation $p_{k}(25n + 24 - k) \equiv 0 \pmod{5}$ for $k \in \{1, 2, \ldots, 24\}$, in connection with the 2-color partition function $p_k(n)$, providing references to existing results for $k \in \{1, 2, 3, 4, 7, 8, 17\}$, simple proofs for $k \in \{5, 10, 15, 20\}$ for the sake of completeness, and counter-examples in all the remaining cases. We also propose an alternative proof in the case of $k = 4$, without using the Rogers-Ramanujan ratio, thereby making the proof considerably simpler compared to the proof by Ahmed, Baruah and Ghosh Dastidar (JNT 2015).

Cylindric partitions and some new $A_2$ Rogers-Ramanujan identities

2020-01-01
Sylvie Corteel , Jehanne Dousse , Ali Kemal Uncu
We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo … We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with quadruple sums, related with work of Andrews, Schilling, and Warnaar.
View PDF Chat

Congruences for 5-regular partitions modulo powers of 5

2016-03-07
Liuquan Wang

Ramanujan-type congruences modulo powers of 5 and 7

2017-09-01
D. Ranganatha

Ramanujans Vermutung über Zerfällungszahlen.

1938-01-01
G. N. Watson

A simple proof of the Ramanujan conjecture for powers of 5.

1981-07-01
Ramanujan conjecture for powers of 5 Abstract.Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is … Ramanujan conjecture for powers of 5 Abstract.Ramanujan conjectured, and G. N. Watson proved, that if n is of a specific form then p(n), the number of partitions of n, is divisible by a high power of 5.In the present note, we establish appropriate generating function formulae, from which the truth of Ramanujan's conjecture, as well as some results of a similar type due to Watson, are shown to follow easily.Furthermore, we derive two new congruences for the partition function.Our proofs are more straightforward than those of Watson and more recent writers and use only classical identities of Euler and Jacobi.
View PDF Chat

Collected Papers of Srinivasa Ramanujan.

1928-06-01
Louis Weisner , G. H. Hardy , P. V. Seshu Aiyar +1 more