Type Congruences Research Articles

Andrews-Beck type congruences modulo arbitrary powers of 5 for 2-colored partitions

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. Chern established some results on weighted rank and crank moments and proved many Andrews–Beck type congruences. Motivated by Andrews and Chern’s work, Lin, Peng and Toh introduced a partition statistic of k-colored partitions [Formula: see text] which counts the total number of parts of the first component in each k-colored partition [Formula: see text] of n with [Formula: see text] congruent to r modulo m and proved many congruences for [Formula: see text]. Very recently, Du and Tang proved a number of Andrews–Beck type congruences for [Formula: see text] and confirmed all conjectures posed by Lin, Peng and Toh. Motivated by their work, we establish the generating functions of [Formula: see text] and prove several families of congruences modulo arbitrary powers of 5 for [Formula: see text]. In particular, we generalize a congruence modulo 5 for [Formula: see text] due to Lin, Peng and Toh.

Converses to Generalized Conway–Gordon Type Congruences

Converses to Generalized Conway–Gordon Type Congruences

Ramanujan type congruences for quotients of Klein forms

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p>19. Canonical bases for modular forms of level 5≤p≤19 are constructed by summing weight one Hecke Eisenstein series of levels 5≤p≤19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ1(p) and Γ(p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ1(p) as a direct sum of subspaces of modular forms for Γ(p) of the form qr/pZ[[q]]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences.

Partitions with parts separated by parity: conjugation, congruences and the mock theta functions

Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions $\omega (q)$ , $\nu (q)$ , and $\psi ^{(3)}(q)$ , introduced by Ramanujan and Watson.

Proofs of two conjectural Andrews–Beck type congruences due to Lin, Peng and Toh

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of [Formula: see text] with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for [Formula: see text]-colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo [Formula: see text] by utilizing some [Formula: see text]-series techniques and the theory of modular forms.

Congruences for Fourier coefficients of eta‐quotients modulo powers of 5, 7, 11, 13, and 17

In this paper, we investigate the Fourier coefficients of the eta‐quotients of the forms where is the Dedekind eta function, , and 17; is a positive integer, and are arbitrary integers. We prove Ramanujan's type congruences for the Fourier coefficients of modulo powers of prime . We recover several results due to Atkin, Garvan, Gordon, Wang, Mestrige, and others. We give few examples, and we establish an improvement for Wang's results related to the 11‐regular partition function.

Proofs of some conjectures of Chan–Mao–Osburn on Beck’s partition statistics

Recently, George Beck introduced two partition statistics NT(m, j, n) and \(M_{\omega }(m,j,n)\), which denote the total number of parts in the partition of n with rank congruent to m modulo j and the total number of ones in the partition of n with crank congruent to m modulo j, respectively. Andrews proved a congruence on NT(m, 5, n) which was conjectured by Beck. Very recently, Chan, Mao and Osburn established a number of Andrews–Beck type congruences and posed several conjectures involving NT(m, j, n) and \(M_{\omega }(m,j,n)\). Some of those conjectures were proved by Chern and Mao. In this paper, we confirm the remaining three conjectures of Chan–Mao–Osburn and two conjectures due to Mao. We also present two new conjectures on \(M_{\omega }(m,j,n)\) and NT(m, j, n).

Andrews-Beck Type Congruences for Overpartitions

We prove Andrews-Beck type congruences for overpartitions concerning the $D$-rank and $M_2$-rank. To prove congruences, we establish the generating function for weighted $D$-rank (respectively, $M_2$-rank) moment of overpartitions and find a connection with the second $D$-rank (respectively, $M_2$-rank) moment for overpartitions.

Combinatorial proofs of the Ramanujan type congruences modulo 3

The partition statistic VR-rank is introduced to give combinatorial proofs of the Ramanujan type congruences mod 3 for certain classes of partition functions.

Proof of a Lin-Peng-Toh's conjecture on an Andrews-Beck type congruence

Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities.

Weighted partition rank and crank moments. III. A list of Andrews–Beck type congruences modulo 5, 7, 11 and 13

Let [Formula: see text] count the total number of parts among partitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text] and let [Formula: see text] count the total appearances of ones among partitions of [Formula: see text] with crank congruent to [Formula: see text] modulo [Formula: see text]. We provide a list of over 70 congruences modulo [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] involving [Formula: see text] and [Formula: see text], which are known as congruences of Andrews–Beck type. Some recent conjectures of Chan, Mao and Osburn are also included in this list.

Weighted generalized crank moments for k-colored partitions and Andrews-Beck type congruences

Recently, Beck studied a new partition statistic which involves counting the total number of parts of a partition with certain rank or crank. Andrews proved two of Beck's conjectures related to ranks. Chern subsequently proved several results involving weighted rank and crank moments and deduced a number of similar Andrews-Beck type congruences. In this paper, we show that some of Chern's results can be explained by a simple combinatorial argument, and extend this approach to the study of k-colored partitions. As a consequence, we derive a large number of new Andrews-Beck type congruences for k-colored partitions.

On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series

Let $p$ be an odd prime and let $n$ be a positive integer with $p\nmid n$. For any positive integer $r$ and $\lambda \in \{1, 2, 3\}$ with $p\nmid \lambda $, we have $$ \sum _{k=0}^{p^{r}n-1}\frac {\left (\frac 12\right )_k}{k!}\cdot \frac {4^k}{\lambda ^

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.

Ramanujan type congruences for quotients of level 7 Klein forms

Klein forms are used to construct generators for the graded algebra of modular forms of level 7. Dissection formulas for the series imply Ramanujan type congruences modulo powers of 7 for a family of generating functions that subsume the counting function for 7-core partitions. The broad class of arithmetic functions considered here enumerate colored partitions by weights determined by parts modulo 7. The method is a prototype for similar analysis of modular forms of level 7 and at other prime levels. As an example of the utility of the dissection method, the paper concludes with a derivation of novel congruences for the number of representations by x2+xy+2y2 in exactly k ways.

On Eight Colour Partitions

In 2013, Baruah and Sarmah, and Xia and Yao independently obtained generating function for the sequences p−8(2n + 1) and p−8(4n + 3), where p−8(n) counts the number of partitions of n in eight colours. In this article, we generalize the identities and as a consequence, establish several Ramanujan type congruences modulo higher powers of 2.

On $k$-restricted overpartitions

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.

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

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 $B_{4, 5}(n)$.

Infinite families of congruences for k-regular overpartitions

Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we find infinite families of congruences modulo 4, 8 and 16 for [Formula: see text] and [Formula: see text] for any [Formula: see text]. Along the way, we obtain several Ramanujan type congruences for [Formula: see text] and [Formula: see text]. We also find infinite families of congruences modulo [Formula: see text] for [Formula: see text].

New congruences for overcubic partition pairs

In this paper, we study congruence properties of overcubic partition pairs. Let $\bar{b}(n)$ denote the number of overcubic partition pairs of $n$. We will establish some new Ramanujan type congruences and several infinite families of congruences modulo powers of $2$ satisfied by $\bar{b}(n)$.