Ramanujan’s Elementary Method in Partition Congruences

Berndt, Bruce C.; Gugg, Chadwick; Kim, Sun

doi:10.1007/978-1-4614-0028-8_2

Cited by 9 publications

(5 citation statements)

References 13 publications

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“…Two classes of partition-theoretic objects that have excited much research are overpartitions, defined in 2004 in [5] and since studied in [6], [7], and [12] among many other papers; and k-colored partitions or multipartitions, which have a longer history, being of interest to see Ramanujan [4] and many authors since: see [2] for a good survey. Precise definitions for these are given below.…”

Section: Introductionmentioning

confidence: 99%

Restricted k-color partitions

Keith

2015

Ramanujan J

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Abstract. We generalize overpartitions to (k, j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. We find connections to divisor sums, the Han/Nekrasov-Okounkov hook length formula and a possible approach to a finitization, and other topics, suggesting that a rich mine of results is available.

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Section: Introductionmentioning

confidence: 99%

Restricted k-color partitions

Keith

2015

Ramanujan J

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“…There exists a third, more complicated relation of this type due to Winquist [6,7], concerning P (X) −10 , but it is not needed for the results of this paper. Such identities of power series with integer coefficients can be reduced modulo a prime q, leading to elegant proofs of otherwise difficult congruences.…”

Section: Introductionmentioning

confidence: 99%

Computer-assisted proofs of congruences for multipartitions and divisor function convolutions, based on methods of differential algebra

Pascadi

2021

Ramanujan J

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This paper provides algebraic proofs for several types of congruences involving the multipartition function and self-convolutions of the divisor function. Our computations use methods of Differential Algebra in Z qZ, implemented in a couple of MAPLE programs available as ancillary files on arXiv [37,38].The first results of the paper are Ramanujan-type congruences of the form p * k (qn + r) ≡q 0 and σ * k (qn + r) ≡q 0, where p(n) and σ(n) are the partition and divisor functions, q > 3 is prime, and * k denotes k th -order self-convolution. We prove all the valid congruences of this form for q ∈ {5, 7, 11}, including the three Ramanujan congruences [1], and a nontrivial one for q = 17. All such multipartition congruences have already been settled in principle up to a numerical verification due to D. Eichhorn and K. Ono [2] via modular forms, but our proofs are purely algebraic. On the other hand, the majority of the divisor function congruences are new results.We then proceed to search for more general congruences modulo small primes, concerning linear combinations of σ * k (pn + r) for different values of k, as well as weighted convolutions of p(n) and σ(n) with polynomial weights. The paper ends with a few corollaries and extensions for the divisor function congruences, including proofs for three conjectures of N. C. Bonciocat.

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“…Ramanujan [31, p. 182] also initiated the study of the function p t (n) for non-zero integer t = 1. We refer to the paper by Berndt, Gugg and Kim [7] for further comments on Ramanujan's study. For some more work on p t (n) for non-zero integer t = 1, see [2,3,9,19,20,24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Families of Congruences for Fractional Partition Functions Modulo Powers of Primes

Baruah,

Das

2020

Preprint

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Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59-80 ( 2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.

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Ramanujan’s Elementary Method in Partition Congruences

Cited by 9 publications

References 13 publications

Restricted k-color partitions

Restricted k-color partitions

Computer-assisted proofs of congruences for multipartitions and divisor function convolutions, based on methods of differential algebra

Families of Congruences for Fractional Partition Functions Modulo Powers of Primes

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