Modular forms and $k$-colored generalized Frobenius partitions

Chan, Heng Huat; Wang, Liuquan; Yang, Yifan

doi:10.1090/tran/7447

Cited by 17 publications

(14 citation statements)

References 42 publications

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“…In [And84a, (4.8)], Andrews defined a generalisation of Frobenius partitions where λ and µ are partitions into distinct parts chosen from {k j : k ∈ N, 1 ≤ j ≤ n}, where k j = k j if and only if k = k and j = j . Their generating function CΦ k (q) has been widely studied from the point of view of modular forms and congruences, see for example [CWY19,Lov00,Sel94].…”

Section: Statement Of Resultsmentioning

confidence: 99%

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Dousse¹,

Konan²

2019

Preprint

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The partition identities of Capparelli and Primc were originally discovered via representation theoretic techniques, and have since then been studied and refined combinatorially, but the question of giving a very broad generalisation remained open. In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra A(1)In this first paper, we focus on combinatorial aspects. We give a n 2 -coloured generalisation of Primc's identity by constructing a n 2 × n 2 matrix of difference conditions, Primc's original identities corresponding to n = 2 and n = 3. While most coloured partition identities in the literature connect partitions with difference conditions to partitions with congruence conditions, in our case, the natural way to generalise these identities is to relate partitions with difference conditions to coloured Frobenius partitions. This gives a very simple expression for the generating function. With a particular specialisation of the colour variables, our generalisation also yields a partition identity with congruence conditions.Then, using a bijection from our new generalisation of Primc's identity, we deduce two families of identities on (n 2 − 1)-coloured partitions which generalise Capparelli's identity, also in terms of coloured Frobenius partitions. The particular case n = 2 is Capparelli's identity and the case n = 3 recovers an identity of Meurman and Primc.In the second paper, we will focus on crystal theoretic aspects. We will show that the difference conditions we defined in our n 2 -coloured generalisation of Primc's identity are actually energy functions for certain A(1) n−1 crystals. We will then use this result to retrieve the Kac-Peterson character formula and derive a new character formula as a sum of infinite products for all the irreducible highest weight A(1) n−1 -modules of level 1.

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Section: Statement Of Resultsmentioning

confidence: 99%

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Dousse¹,

Konan²

2019

Preprint

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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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“…Very recently in [3], Chan, Wang and Yan have discovered the following more general relationships between cφ p (n) and P(n). Below, noting that the Dedekind eta function is defined by η(z) = q 1/24 (q; q) ∞ , we restate the main aspects of their Theorem 4.1.…”

Section: Introductionmentioning

confidence: 99%

N-colored generalized Frobenius partitions: generalized Kolitsch identities

Aygin

Nguyen²

2022

Can. J. Math.-J. Can. Math.

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Let N ≥ 1 be squarefree with (N , 6) = 1. Let cφ 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. 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 cφ N (n) in terms of the classical partition function P(n).

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Modular forms and $k$-colored generalized Frobenius partitions

Cited by 17 publications

References 42 publications

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

Generalisations of Capparelli's and Primc's identities, I: coloured Frobenius partitions and combinatorial proofs

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

N-colored generalized Frobenius partitions: generalized Kolitsch identities

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