Congruence properties for a certain kind of partition functions

Cui, Su-Ping; Gu, Nancy S. S.; Huang, Anthony X.

doi:10.1016/j.aim.2015.12.014

Cited by 10 publications

(10 citation statements)

References 28 publications

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“…In fact, this turns out to be a particularly interesting case, since we already know that Ramanujan congruences exist modulo 2, but Theorem 1 does not guarantee that there are infinitely many (non-Ramanujan) congruences. However, using the fact that (1 − x 2 ) 2 ≡ (1 − x) 4 (mod 4), we have that…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Congruences for modular forms and generalized Frobenius partitions

Jameson

Wieczorek

2019

Ramanujan J

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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 [3]. In particular, we prove that there are infinitely many congruences for cφ k (n) modulo ℓ, where gcd(ℓ, 6k) = 1, and we also prove results on the parity of cφ k (n). Along the way, we prove results regarding the parity of coefficients of weakly holomorphic modular forms which generalize work of Ono [15].

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…4. The parity of cφ 2 (n) 4 Recall that we defined cφ 2 (n) := cφ 2 (n) − p(n). In fact, this is a special case of a function cφ k (n), which was defined combinatorially by Kolitsch [9,10], who also found congruences for this function.…”

Section: 2mentioning

confidence: 99%

Congruences for modular forms and generalized Frobenius partitions

Jameson

Wieczorek

2019

Ramanujan J

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“…While there exists an extensive literature on the subject of congruences satisfied by generalized Frobenius partition functions, our focus in this note will be on parity results. We highlight here that a number of authors have proven congruence results with even moduli for these functions; see, for example, the work of Andrews [1, Theorem 10.2], Baruah and Sarmah [2,3], Chan, Wang, and Yang [4], Cui and Gu, [5], Cui, Gu, and Huang [6], and Jameson and Wieczorek [9] where specific congruence results with even moduli are proved. Several additional papers involving congruence results for generalized Frobenius partitions, but with odd moduli, also appear in the literature.…”

Section: Introductionmentioning

confidence: 93%

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

Andrews¹,

Sellers²,

Soufan³

2022

Preprint

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In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, φ k (n) and cφ 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 infinite family of values of k such that φ 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 ℓ, all primes p ≥ 5, and all values r, 0 < r < p, such that 24r + 1 is a quadratic nonresidue modulo p, φ pℓ−1 (pn + r) ≡ 0 (mod 2) for all n ≥ 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.

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“…Motivated by Ramanujan's work, the arithmetic properties of partitions with certain restrictions have received a great deal of attention. Recently, Cui et al [3] established some congruence properties for a certain kind of partition function a(n) which satisfies ∞ n=0 a(n)q n ≡ (q; q) k ∞ (mod m), where k is a positive integer with 1 ≤ k ≤ 24 and m ∈ {2, 3} in light of the modular equations of fifth and seventh order.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we prove congruences modulo 16 for a φ (n) based on Theorems 1.1-1. 3. In order to state congruences modulo 16 for a φ (n), define π 1 (p) := p 9 3(p 2 − 1) 8 + (−1) (p−1)(p−19)/8 p 3 and…”

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confidence: 99%

Newman's identities, lucas sequences and congruences for certain partition functions

Xia¹

2020

Proceedings of the Edinburgh Mathematical Society

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Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] and for k ≥ 0, \[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] where spt(n) denotes Andrews's smallest parts function.

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Congruence properties for a certain kind of partition functions

Cited by 10 publications

References 28 publications

Congruences for modular forms and generalized Frobenius partitions

Congruences for modular forms and generalized Frobenius partitions

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

Newman's identities, lucas sequences and congruences for certain partition functions

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