Some congruences for modulus 13 related to partition generating function

Bilgici, Göksal; Ekin, Ali Bülent

doi:10.1007/s11139-013-9537-4

Cited by 6 publications

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“…By [5, (51)], with a small correction to the q 7 term, we have that E(q) 10 As noted in [12] and [5], one can use [13, (LVII2)], to get that Again it would appear we could do even perhaps, calculations suggest that 11q 5 P (2)P (3)P (4)P (6) + 6q 5 P (1)P (4)P (5)P (6) + 5q 18 P (1)P (2)P (3)P (5) ≡ E(13) 10 E(169) 2 , however this form is not as useful for our calculations. We multiply (3.9) by (3.11), collect terms, and reduce modulo 13, to find that , A ′ 13,7 (q 13 ) = + 12P (4) 2 P (3) 2 P (6) 2 q 13 P (1) 2 P (5) To finish the proof of (1.7) we must show that A ′ 13 ≡ A 13 (mod 13).…”

Section: Proof Of Theorem 12mentioning

confidence: 98%

Two Partition Functions with Congruences Modulo 3,5,7, and 13

Jennings-Shaffer

2017

Ann. Comb.

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Abstract. We introduce two new integer partition functions, both of which are the number of partition quadruples of n with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo 3, 5, 7, and 13 by use of generalized Lambert series identities and q-series techniques. IntroductionWe recall a partition of a positive integer n is a non-increasing sequence of positive integers that sum to n. For example, the partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1. We let p(n) denote the number of partitions of n. The function p(n) satisfies the well known congruences of Ramanujan p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), and p(11n + 6) ≡ 0 (mod 11). In this article we will consider two partition quadruples of n. We say a quadruple (π 1 , π 2 , π 3 , π 4 ) of partitions is a partition quadruple of n is altogether the parts of π 1 , π 2 , π 3 , and π 4 sum to n.For a partition π, we let s(π) denote the smallest part of π and ℓ(π) denote the largest part of π. We use the conventions that the empty partition has smallest part ∞ and largest part 0. We let u(n) denote the number of partition quadruples (π 1 , π 2 , π 3 , π 4 ) of n such that π 1 is non-empty,, and ℓ(π 4 ) ≤ 2s(π 1 ). We let v(n) denote the number of partition quadruples (π 1 , π 2 , π 3 , π 4 ) of n such that the smallest part of π 1 appears at least twice, s(π 1 ) ≤ s(π 2 ), s(π 1 ) ≤ s(π 3 ), s(π 1 ) ≤ s(π 4 ), and ℓ(π 4 ) ≤ 2s(π 1 ).We use the standard product notation,By summing according to n being the smallest part of a partition, one easily deduces that a generating function for p(n) is given bySimilarly, by summing according to n being the smallest part of the partition π 1 , we find that generating functions for u(n) and v(n) are given by(q n ; q) ∞ (q n ; q) ∞ (q n ; q) ∞ (q n ; q) n+1 .The main result of this article are the following congruences for u(n) and v(n).

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Section: Proof Of Theorem 12mentioning

confidence: 98%

Two Partition Functions with Congruences Modulo 3,5,7, and 13

Jennings-Shaffer

2017

Ann. Comb.

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“…Using the following lemma which is given by the authors in [2], we can determine the g s in terms of P (a) .…”

Section: Preliminariesmentioning

confidence: 99%

“…We use the following lemma which is given by the authors in [2] to obtain the congruence properties of components.…”

Section: We Definementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In a recent paper [2], the authors obtained 13-dissection for F and some congruences for the components F (k,13) where 0 ≤ k ≤ 12. In this paper, by following the same course in [2], we obtain 11-dissection for F. The 11-dissection for F may appear for the first time. After obtaining the components, using a q-equivalence given in [2], we obtain the congruences for the components F (k,11) (mod 11) given by Atkin and Swinnerton-Dyer in [1].…”

Section: Introductionmentioning

confidence: 99%

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11-dissection and modulo 11 congruences properties for partition generating function

Bilgici

Ekin

2014

ijcms

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In a recent paper, we give 13-dissection and some congruences for modulo 13 for the partition generating function (1 − q r ) −1 by using a method of Kolberg. In this paper, by following similar course, we develop an algoritmic approach and give 11-dissection for the partition generating function (1 − q r ) −1 . Then we re-obtain the congruences given by Atkin and Swinnerton-Dyer. Mathematics Subject Classification: 11P83

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