Families Of Congruences Modulo Research Articles

Congruences for k-elongated plane partition diamonds

In the eleventh paper in the series on MacMahon’s partition analysis, Andrews and Paule introduced the [Formula: see text]-elongated partition diamonds. Recently, they revisited the topic. Let [Formula: see text] count the partitions obtained by adding the links of the [Formula: see text]-elongated plane partition diamonds of length [Formula: see text]. Andrews and Paule obtained several generating functions and congruences for [Formula: see text], [Formula: see text], and [Formula: see text]. Da Silva et al. further found many congruences modulo 4, 5, 7, 8, 9, and 11 for [Formula: see text]. In this paper, we extend some individual congruences found by Andrews and Paule and da Silva et al. to their respective families as well as find new families of congruences for [Formula: see text]. We also present a refinement in an existence result for congruences of [Formula: see text] found by da Silva et al. and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64, and 128.

A conjecture of Baruah and Begum on the smallest parts function of restricted overpartitions

In 2017, Andrews, Dixit, Schultz and Yee introduced the function $$\overline{\text {spt}}_\omega (n)$$ , which denotes the number of smallest parts in the overpartitions of n in which the smallest part is always overlined and all odd parts are less than twice the smallest part. Recently, Baruah and Begum established several internal congruences and congruences modulo small powers of 5 satisfied by $$\overline{\text {spt}}_\omega (n)$$ . Moreover, they conjectured a family of internal congruences modulo all powers of 5 and two families of congruences modulo all even powers of 5. In this paper, we confirm the three families of congruences due to Baruah and Begum.

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.

Parity results for broken 11-diamond partitions

Abstract Recently, Dai proved new infinite families of congruences modulo 2 for broken 11-diamond partition functions by using Hecke operators. In this note, we establish new parity results for broken 11-diamond partition functions. In particular, we generalize the congruences due to Dai by utilizing an identity due to Newman. Furthermore, we prove some strange congruences modulo 2 for broken 11-diamond partition functions. For example, we prove that if p is a prime with p ≠ 23 and Δ11(2p + 1) ≡ 1 (mod 2), then for any k ≥ 0, Δ11(2p3k+3 + 1) ≡ 1 (mod 2), where Δ11(n) is the number of broken 11-diamond partitions of n.

Infinite families of congruences modulo 5 and 7 for the cubic partition function

AbstractIn 2010, Hei-Chi Chan introduced the cubic partition function a(n) in connection with Ramanujan's cubic continued fraction. Chen and Lin, and Ahmed, Baruah and Dastidar proved that a(25n + 22) ≡ 0 (mod 5) for n ⩾ 0. In this paper, we prove several infinite families of congruences modulo 5 and 7 for a(n). Our results generalize the congruence a(25n + 22) ≡ 0 (mod 5) and four congruences modulo 7 for a(n) due to Chen and Lin. Moreover, we present some non-standard congruences modulo 5 for a(n) by using an identity of Newman. For example, we prove that $a((({15\times 17^{3\alpha }+1})/{8})) \equiv 3^{\alpha +1} \ ({\rm mod}\ 5)$ for α ⩾ 0.

Arithmetic properties for two restricted partitions modulo powers of 5

Let pk,3(n) count the number of 2-color partition triples of n where one of the colors appears only in parts that are multiples of k and Bk,ℓ(n) denote the number of (k, ℓ)-regular bipartitions of n. In this paper, we prove two infinite families of congruences modulo 5 for p5,3(n), three infinite families of congruences modulo powers of 5 for p25,3(n), and six infinite families of congruences modulo powers of 5 for B5,25(n). For instance, for any integers n ≥ 0 and α ≥ 1, we haveand

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].

Congruences modulo 7 for broken 3-diamond partitions

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Wang and Yao, and Xia proved several infinite families of congruences modulo 7 for \(\Delta _3(n)\) by using theta function identities. In this paper, we give a new proof of one result of Wang and Yao, and find three new infinite families of congruences modulo 7 for \(\Delta _3(n)\).

Arithmetic properties of 7-regular partitions

Let $$b_{\ell }(n)$$ denote the number of $$\ell $$ -regular partitions of n. By employing the modular equation of seventh order, we establish the following congruence for $$b_{7}(n)$$ modulo powers of 7: for $$n\ge 0$$ and $$j\ge 1$$ , $$\begin{aligned} b_{7}\left( 7^{2j-1}n+\frac{3\cdot 7^{2j}-1}{4}\right) \equiv 0 \pmod {7^j}. \end{aligned}$$ We also find some infinite families of congruences modulo 2 and 7 satisfied by $$b_{7}(n)$$ .

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 , , .

Infinite families of congruences modulo 5 and 9 for overpartitions

Infinite families of congruences modulo 5 and 9 for overpartitions

On $$(\ell , m)$$ ( ℓ , m ) -regular partitions with distinct parts

Let $$a_{\ell ,m}(n)$$ denote the number of $$(\ell ,m)$$ -regular partitions of a positive integer n into distinct parts, where $$\ell $$ and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for $$a_{3,5}(n)$$ . For example, $$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$ where $$\alpha , \beta \ge 0$$ .

Ramanujan-type congruences modulo 8 for 7- and 23-core partition functions

Recently, Radu and Sellers proved numerous congruences modulo powers of 2 for \( (2k+1)\)-core partition functions by employing the theory of modular forms. In this paper, employing Ramanujan’s theta function identities, we prove many infinite families of congruences modulo 8 for 7-core partition function. Our results generalize the congruences modulo 8 for 7-core partition function discovered by Radu and Sellers. Furthermore, we present new proofs of congruences modulo 8 for 23-core partition function. These congruences were first proved by Radu and Sellers.

Congruences modulo 4 for broken k-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let $$\Delta _k(n)$$ denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, a number of parity results satisfied by $$\Delta _k(n)$$ for small values of k have been proved by Radu and Sellers and others. However, congruences modulo 4 for $$\Delta _k(n)$$ are unknown. In this paper, we will prove five congruences modulo 4 for $$\Delta _5(n)$$ , four infinite families of congruences modulo 4 for $$\Delta _7(n)$$ and one congruence modulo 4 for $$\Delta _{11}(n)$$ by employing theta function identities. Furthermore, we will prove a new parity result for $$\Delta _2(n)$$ .

Newman’s identity and infinite families of congruences modulo 7 for broken 3-diamond partitions

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu presented some conjectures on congruences modulo 7 for \(\Delta _3(n)\) which were proved by Jameson and Xiong based on the theory of modular forms. Very recently, Xia proved several infinite families of congruences modulo 7 for \(\Delta _3(n)\) using theta function identities. In this paper, many new infinite families of congruences modulo 7 for \(\Delta _3(n)\) are derived based on an identity of Newman and the (p; k)-parametrization of theta functions due to Alaca, Alaca and Williams. In particular, some non-standard congruences modulo 7 for \(\Delta _3(n)\) are deduced. For example, we prove that for \(\alpha \ge 0\), \(\Delta _3\left( \frac{14\times 757^{\alpha }+1}{3}\right) \equiv 6 -\alpha \ (\mathrm{mod}\ 7)\).

Note on the parity of broken 11-diamond partitions

In this paper, we prove new infinite families of congruences modulo 2 for broken 11-diamond partitions by using Hecke operators.

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Infinite families of congruences modulo 7 for broken 3-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu conjectured that \(\Delta _3(343n+82)\equiv \Delta _3(343n+278)\equiv \Delta _3(343n+327)\equiv 0\ (\mathrm{mod} \ 7)\). Jameson confirmed this conjecture and proved that \(\Delta _3(343n+229)\equiv 0 \ (\mathrm{mod} \ 7)\) by using the theory of modular forms. In this paper, we prove several infinite families of Ramanujan-type congruences modulo 7 for \(\Delta _3(n)\) by establishing a recurrence relation for a sequence related to \(\Delta _3(7n+5)\). In the process, we also give new proofs of the four congruences due to Paule and Radu, and Jameson.

Arithmetic properties of bipartitions with 3-cores

Recently, Lin discovered several nice congruences modulo 4, 5, 7 and 8 for \(A_3(n)\), where \(A_3(n)\) is the number of bipartitions with 3-cores of \(n\). For example, Lin proved that for all \(\alpha \ge 0\) and \(n\ge 0\), \(A_3(16^{\alpha +1}n+\frac{2^{4\alpha +3}-2}{3})\equiv 0 \ (\mathrm{mod }\ 5)\). Yao also established several infinite families of congruences modulo 3 and 9 for \(A_9(n)\). In this paper, several infinite families of congruences modulo 4, 8 and \(\frac{4^k-1}{3}\ (k\ge 2)\) for \(A_3(n)\) are established. We generalize some results due to Lin and Yao. For example, we prove that for \(n\ge 0, \ \alpha \ge 0\) and \(k\ge 2\), \( A_3\left( 4^{k(\alpha +1)} n+\frac{2^{2k(\alpha +1)-1}-2}{3}\right) \equiv 0 \ (\mathrm{mod\ } \frac{4^{k}-1}{3}) \).

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$