Generalized Frobenius Partitions Research Articles

Some congruences for 12-coloured generalized Frobenius partitions

Abstract In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$ , the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$ . In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$ .

The method of constant terms and k-colored generalized Frobenius partitions

In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let cϕk(n) denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any n≥0, cϕ2(5n+3)≡0(mod5). Since then, many scholars subsequently considered congruence properties of various k-colored generalized Frobenius partition functions, typically with a small number of colors.In 2019, Chan, Wang and Yang systematically studied arithmetic properties of CΦk(q) with 2≤k≤17 by employing the theory of modular forms, where CΦk(q) denotes the generating function of cϕk(n). We notice that many coefficients in the expressions of CΦk(q) are not integers. In this paper, we first observe that CΦk(q) is related to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting and computing the constant terms of these bivariable identities, we establish the expressions of CΦk(q) with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by cϕk(n), where k is allowed to grow arbitrary large.

ON THE PARITY OF THE GENERALISED FROBENIUS PARTITION FUNCTIONS

AbstractAndrews [Generalized Frobenius Partitions, Memoirs of the American Mathematical Society, 301 (American Mathematical Society, Providence, RI, 1984)] defined two families of functions, $\phi _k(n)$ and $c\phi _k(n),$ enumerating two types of combinatorial objects which he called generalised Frobenius partitions. Andrews proved a number of Ramanujan-like congruences satisfied by specific functions within these two families. Numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ Our goal is to identify an infinite family of values of k such that $\phi _k(n)$ is even for all n in a specific arithmetic progression; in particular, we 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 $p,$ $$ \begin{align*} \phi_{p\ell-1}(pn+r) \equiv 0 \pmod{2} \end{align*} $$ 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 generalised Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

Congruences for generalized Frobenius partitions with 6 colors

We establish the generating function for cϕ6¯(n), the number of generalized Frobenius partitions of n with 6 colors whose order is 6 under cyclic permutation of the 6 colors. Furthermore, we find some congruences for cϕ6¯(n) modulo 24.

Generalized Frobenius partitions, Motzkin paths, and Jacobi forms

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which yields explicit formulas for the generalized Frobenius partition generating functions in terms of infinite q-products. In particular, we show that specific examples of our result easily reestablish previously known formulas, and we describe new congruences, both conjectural and proven, in additional cases. The modular structure of Jacobi forms indicates that all of the coefficients of the forms are of interest. We give a combinatorial definition of these “companion series” and explore their combinatorics via the counting of Motzkin paths.

N-colored generalized Frobenius partitions: generalized Kolitsch identities

AbstractLet $N\geq 1$ be squarefree with $(N,6)=1$ . Let $c\phi _N(n)$ denote the number of N-colored generalized Frobenius partitions of n introduced by Andrews in 1984, and $P(n)$ denote the number of partitions of n. We prove $$ \begin{align*}c\phi_N(n)= \sum_{d \mid N} N/d \cdot P\left( \frac{ N}{d^2}n - \frac{N^2-d^2}{24d^2} \right) + b(n),\end{align*} $$ where $C(z) := (q;q)^N_\infty \sum _{n=1}^{\infty } b(n) q^n$ is a cusp form in $S_{(N-1)/2} (\Gamma _0(N),\chi _N)$ . This extends and strengthens earlier results of Kolitsch and Chan–Wang–Yan treating the case when N is a prime. As an immediate application, we obtain an asymptotic formula for $c\phi _N(n)$ in terms of the classical partition function $P(n)$ .

Incongruences for modular forms and applications to partition functions

The study of arithmetic properties of coefficients of modular forms f(τ)=∑a(n)qn has a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of C.-S. Radu, S. Ahlgren, B. Kim, N. Andersen, and S. Löbrich have employed the q-expansion theory of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we apply the method to a large class of modular forms, and in particular to several noteworthy examples, including generalized Frobenius partitions and the two mock theta functions f(q) and ω(q).

Congruences for modular forms and generalized Frobenius partitions

The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions defined by Andrews (Mem Am Math Soc 49(301):iv+44, 1984). In particular, we prove that there are infinitely many congruences for $$c\phi _k(n)$$ modulo $$\ell ,$$ where $$\gcd (\ell ,6k)=1,$$ and we also prove results on the parity of $$c\phi _k(n).$$ Along the way, we prove results regarding the parity of coefficients of weakly holomorphic modular forms which generalize work of Ono (J Reine Angew Math 472:1–15, 1996).

Congruences modulo powers of 2 for generalized Frobenius partitions with six colors

A generalized Frobenius partition of [Formula: see text] with [Formula: see text] colors is a two-rowed array [Formula: see text] where [Formula: see text], and the integer entries are taken from [Formula: see text] distinct copies of the non-negative integers distinguished by color, and the rows are ordered first by size and then by color with no two consecutive like entries in any row. Let [Formula: see text] denote the number of this kind of partitions of [Formula: see text] with [Formula: see text] colors. In this paper, we establish some congruences modulo powers of 2 for [Formula: see text].

Modular forms and $k$-colored generalized Frobenius partitions

Let $k$ and $n$ be positive integers. Let $c\phi_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$ and $\mathrm{C}\Phi_k(q)$ be the generating function of $c\phi_{k}(n)$. In this article, we study $\mathrm{C}\Phi_k(q)$ using the theory of modular forms and discover new surprising properties of $\mathrm{C}\Phi_k(q)$.

Congruences modulo powers of 3 for 3- and 9-colored generalized Frobenius partitions

Let cϕk(n) be the number of k-colored generalized Frobenius partitions of n. We establish some infinite families of congruences for cϕ3(n) and cϕ9(n) modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for k≥3 and n≥0, we prove that cϕ3(32kn+7⋅32k+18)≡0(mod34k+5).We give two different proofs to the congruences satisfied by cϕ9(n). One of the proofs uses a relation between cϕ9(n) and cϕ3(n) due to Kolitsch, for which we provide a new proof in this paper.

On certain weighted 7-colored partitions

Inspired by Andrews’ 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.

Split (n + t)-color partitions and 2-color F-partitions

Andrews [Generalized Frobenius partitions. Memoirs of the American Math. Soc., 301:1{44, 1984] defined the two classes of generalized F-partitions: F-partitions and k-color F-partitions. For many q-series and Rogers-Ramanujan type identities, the bijections are established between F-partitions and (n + t)-color partitions. Recently (n + t)-color partitions have been extended to split (n+t)-color partitions by Agarwal and Sood [Split (n+t)-color partitions and Gordon-McIntosh eight order mock theta functions. Electron. J. Comb., 21(2):#P2.46, 2014]. The purpose of this paper is to study the k-color F-partitions as a combinatorial tool. The paper includes combinatorial proofs and bijections between split (n + t)-color partitions and 2-color F-partitions for some generalized q-series. Our results further give rise to innate three-way combinatorial identities in conjunction with some Rogers-Ramanujan type identities for some particular cases.

CONGRUENCES MODULO 5 AND 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.

A unified proof of Ramanujan-type congruences modulo 5 for 4-colored generalized Frobenius partitions

Inspired by the recent work of Hirschhorn and Sellers, in this paper we present a unified proof of Ramanujan-type congruences modulo 5 for cϕ4(5n+1).

Enumeration of diagonally colored Young diagrams

In this note we give a new proof of a closed formula for the multivariable generating series of diagonally colored Young diagrams. This series also describes the Euler characteristics of certain Nakajima quiver varieties. Our proof is a direct combinatorial argument, based on Andrews’ work on generalized Frobenius partitions. We also obtain representations of these series in some particular cases as infinite products.

Congruences involving arithmetic progressions for weakly holomorphic modular forms

In this paper, we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f=∑a(n)qn of weight k on Γ1(N) having infinitely many congruences of the forma(ℓn+β)≡0(modℓ), where ℓ is a prime and β is an integer in {0,1,⋯,ℓ−1}. These are similar to congruences for the partition function investigated by Ramanujan. Furthermore, we characterize linear combinations of Shimura theta functions with odd characters in terms of these congruences. As an application of our main theorem, we consider a generalization of Newman's conjecture for weakly holomorphic modular forms on Γ0(N) with real Dirichlet characters. Finally we use these results to study arithmetic properties of colored partitions and generalized Frobenius partitions.

An unexpected Ramanujan-type congruence modulo 7 for 4-colored generalized Frobenius partitions

In this paper, we present an unexpected Ramanujan-type congruence modulo 7 for \(c\phi _4(n)\), which denotes the number of generalized Frobenius partitions of n with 4 colors. This work extends the recent work of Lin on \(c\phi _4\) modulo 7.

Congruence properties for a certain kind of partition functions

In light of the modular equations of fifth and seventh order, we derive some congruence properties for a certain kind of partition functions a(n) which satisfy ∑n=0∞a(n)qn≡(q;q)∞k(modm), where k is a positive integer with 1≤k≤24 and m=2,3. In view of these properties, we obtain many infinite families of congruences for cϕk(n), the number of generalized Frobenius partitions of n with k colors, and cϕk‾(n), the number of generalized Frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors. Meanwhile, we also apply the main theorems to some other kinds of partition functions.

Congruences modulo powers of 3 for generalized Frobenius partitions with six colors

Congruences modulo powers of 3 for generalized Frobenius partitions with six colors