Congruences modulo prime for fractional colour partition function

In the present work, we discover some new congruences modulo 5 for pr(n), the general partition function by restricting r to some sequence of negative integers. Our emphasis throughout this paper is to exhibit the use of q-identities to generate the congruences for pr(n).

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 $$\Delta _2(n)$$ .

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial result of Ramanujan’s congruence modulo 11. In 1988, Andrews and Garvan stated such functions and described the celebrated result that the crank simultaneously explains the three Ramanujan congruences modulo 5, 7 and 11. Dyson wrote the article, titled Some Guesses in the theory of partitions, for Eureka, the undergraduate mathematics journal of Cambridge. He discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famous congruences for P (n), the number of partitions of n indeed, Ramanujan’s formulas lay unread until 1976 when Dyson found In the Trainty College Library of Cambridge University among papers from the estate of the late G.N.Watson. In 1986, F.Garvan wrote his Pennsylvania state Ph.D. Thesis Precisely on the formulas of Ramanujan relative to the crank. In view of this theoretical description, the story of the crank is a long romantic tale and the crank functions are intimately connected to all partitions congruences. In 2005, Mahlburg stated that the crank functions themselves obey Ramanujan type congruences.

In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

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.

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.

CHAPTER 18 - NETWORKS WITH MULTI-POSITION CONTACTS

In 1988, Andrews and Garvan introduced the partition statistic “crank” in order to give combinatorial interpretation for Ramanujan’s celebrated partition congruence modulo 11. In 2009, Choi, Kang and Lovejoy established congruences modulo powers of 5 for the crank parity function pC (n) by utilizing the theory of modular forms, where pC (n) denotes the difference between the number of partitions of n with even crank and the number of partitions of n with odd crank. In this paper, we provide a completely elementary proof of this congruence family. Moreover, we also prove several new congruences modulo small powers of 5.

Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for PD$_t(n)$ and PDO$_t(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for PD$_t(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2.

Bacher and de la Harpe ( arxiv:1603.07943 , 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and $$p(n)_\mathbf{e }$$ , a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe ( arxiv:1603.07943 , 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function $$p_{k}(n)$$ , we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes $$\ell \ge 5$$ .

In 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.

Each of these two isolated pages has a connection with Ramanujan’s famous paper in which he gives the first proofs of the congruences p(5n+4)≡0 (mod 5) and p(7n+5)≡0 (mod 7). One of Ramanujan’s proofs hinges upon the beautiful identity $$\sum_{n=0}^{\infty}p(5n+4)q^n = 5 \frac{(q^5;q^5)_{\i}^5}{(q;q)_{\infty}^6}, \qquad |q|<1, $$ which is given on page 189. We provide a more detailed rendition of the proof given by Ramanujan, as well as a similarly beautiful identity yielding the congruence p(7n+5)≡0 (mod 7). On both pages, Ramanujan examines the more general partition function p r (n) defined by $$ \frac{1}{(q;q)_{\infty}^r}=\sum_{n=0}^{\infty}p_r(n)q^n, \qquad |q|<1. $$ In particular, he states new congruences for p r (n).

Potts model partition functions for self-dual families of strip graphs

Kathrin Bringmann (Mathematisches Institut, Universität Köln, Weyertal 86-90, D-50931 Köln, Germany)&#13;\nBen Kane (Wiskunde Afdeling, Radboud Universiteit, Postbus 9010, 6500 GL, Nijmegen, Netherlands)

In 2018, Andrews defined a partition function [Formula: see text] which denotes the number of partitions of n in which every even part is less than each odd part and only the largest even part, if any, appears an odd number of times. He also proved that [Formula: see text]. In order to explain the congruence modulo 5, Andrews introduced the even–odd crank for partitions with even parts below odd parts to be the largest even part minus the number of odd parts. Recently, Fu and Tang defined a partition statistic [Formula: see text] which denotes the total number of odd parts of partitions counted by [Formula: see text] with even–odd cranks congruent to r modulo m and proved two Andrews–Beck-type congruences for [Formula: see text]. In this paper, we prove some identities involving [Formula: see text] which are analogous to Ramanujan’s “most beautiful identity”. From those identities, we can get Fu–Tang’s congruences and new Andrews–Beck-type congruences modulo 4 for [Formula: see text].