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formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

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Some congruences for 3-component multipartitions

Tao Yan Zhao
  • Corresponding author
  • Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China
  • Email:
/ Lily J. Jin
  • School of Mathematics, Nanjing Normal University, Taizhou College, Jiangsu, Taizhou 225300, China
  • Email:
/ C. Gu
  • Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China
  • Email:
Published Online: 2016-10-16 | DOI: https://doi.org/10.1515/math-2016-0067

Abstract

Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).

Keywords: Congruences; Multipartitions; Theta functions

MSC 2010: 11P83; 05A17

1 Introduction

The objective of this paper is to prove several congruences modulo powers of 3 for 3-component multipartitions by employing theta function identities.

Recall that the k-tuple λ = (λ1, λ2,…, λk) of integers λi (i = 1,2,…, k) is a partition of the natural number n if n=i=1kλi and λ1λ2 ≥ ··· ≥ λk > 0. The partition function p(n) is defined to be the number of partitions of n. A multipartition of n with r-components, as called by Andrews [1], also referred to as an r-colored partition, see, for example [2, 3], is an r-tuple λ = (λ(1), λ(2), ··· λ(r)) of partitions whose weights sum to n. Let pr(n) denote the number of r-component multipartitions of n. Multipartitions arise in combinatorics, representation theory and physics, see, for example Bouwknegt [4] and Fayers [5]. As usual, set pr(0) = 1. The generating function of pr(n) is n=0prnqn=1q;qr,(1)

where (a;q)=n=11aqn.(2)

A number of congruences satisfied by pr(n) were discovered, see, for example Andrews [1], Eichhorn and Ono [3], Atkin [6], Baruah and Ojah [7], Boylan [8], Cheema and Haskell [9], Gordon [10], Kiming and Olsson [11], Newman [12], Sinick [13], Treneer [14], Xia [15] and Yao [16]. Recently, Baruah and Ojah [7] established a 3-dissection formula for the generating function for p3(n) and proved that for n ≥ 0, p39n+50mod33(3)

and p39n+80mod34.(4)

In this paper, we prove several congruences modulo powers of 3 for p3(n). The main results of this paper can be stated as follows.

Theorem 1.1: For n > 0, p327n+170mod35,(5) p327n+260mod37,(6) p381n+440mod36,(7) p381n+710mod37,(8) p3243n+1520mod38,(9) p3243n+2330mod310,(10) p3729n+3950mod39,(11) p3729n+6380mod310.(12)

2 Some lemmas

In this section, we collect three lemmas which are needed to prove the main results of this paper.

Lemma 2.1: The following 3-dissection formulas are true: aq=aq3+6qq9;q93q3;q3(13) and bq=aq33qq9;q93q3;q3(14) where a(q) and b(q) are defined by aq=m,n=qm2+mn+n2(15) and bq=m,n=ωmnqm2+mn+n2,ω=exp2πi/3.(16) Lemma 2.1 was proved by Borwein, Borwein and Garvan [18].

Lemma 2.2: We have bq=q;q3q3;q3.(17) Lemma 2.2 was also proved by Borwein, Borwein and Garvan [18].

Lemma 2.3: The following 3-dissection formula holds: 1q;q3=q9;q93q3;q310a2q3+3qaq3q9;q93q3;q3+9q2q9;q96q3;q32.(18)

Remark: Baruah and Ojah [7] also deduced the following 3-dissection formula for 1q;q3, which is different from (18): 1(q;q)3=(q9;q9)9(q3;q3)121w2(q3)+3qw(q3)+9q2+8q3w(q3)+12q4w2(q3)+16q6w4(q3),(23) where w(q)=(q;q)(q6;q6)3(q2;q2)(q3;q3)3.(24)

3 Proof of Theorem 1.1

Setting r = 3 in (1) and using (18), we obtain n=0p3(n)qn=(q9;q9)3(q3;q3)10a2(q3)+3qa(q3)(q9;q9)3(q3;q3)+9q2(q9;q9)6(q3;q3)2,(25)

which yields n=0p3(3n+2)qn=9q3;q39q;q12.(26)

Substituting (18) into (26), we deduce that n=0p33n+2qn=9q9;q912q3;q331a2q3+3qaq3q9;q93q3;q3+9q2(q9;q9)6(q3;q3)249q9;q912q3;q331a8q3+108qq9;q915q3;q332a7q3+810q2q9;q918q3;q333a6q3+3888q3q9;q921q3;q334a5q3+13851q4q9;q924q3;q335a4q3+34992q5q9;q927q3;q336a3q3+6561q6q9;q930q3;q337a2q3+19683q7q9;q933q3;q338aq3mod310,(27)

which yields n=0p39n+8qn810q3;q318q;q33a6q+34992qq3;q327q;q36a3qmod310.(28)

Substituting (13) and (18) into (28), we see that n=0p39n+8qn810q9;q933q3;q392a2q3+3qq9;q93q3;q3aq3+9q2q9;q96q3;q3211×aq3+6qq9;q93q3;q36+34992qq9;q936q3;q393a2q3+3qq9;q93q3;q3aq3+9q2q9;q96q3;q3212×aq3+6qq9;q93q3;q33810q9;q933q3;q392a28q3+31833qq9;q936q3;q393a27q3+50301q2q9;q939q3;q394a26q3+52488q3q9;q942q3;q395a25q3+39366q4q9;q945q3;q396a24q3mod310,(29)

which implies that n=0p327n+17qn131×35q3;q336q;q93a27q+2×39qq3;q345q;q96a24qmod310(30)

and n=0p327n+26qn23×37q3;q339q;q94a26qmod310.(31)

Congruences (5) and (6) follow from (30) and (31).

Substituting (13) and (18) into (30), we get n=0p327n+17qn131×35q9;q993q3;q3274a2q3+3qq9;q93q3;q3aq3+9q2q9;q96q3;q3231×aq3+6qq9;q93q3;q327+2×39qq9;q996q3;q3275a2q3+3qq9;q93q3;q3aq3+9q2q9;q96q3;q3232×aq3+6qq9;q93q3;q32431833q9;q993q3;q3274a89q3+8019qq9;q996q3;q3275a88q3+30618q2q9;q999q3;q3276a87q3+52488q3q9;q9102q3;q3277a86q3+39366q4q9;q9105q3;q3278a85q3mod310.(32)

It follows from (32) that n=0p381n+44qn11×36q3;q396q;q275a88q+2×39qq3;q3105q;q278a85qmod310(33)

and n=0p381n+71qn14×37q3;q399q;q276a87qmod310.(34)

Congruences (7) and (8) follow from (33) and (34).

By the binomial theorem, q3;q390q;q2701mod27.(35)

Combining (34) and (35), we have n=0p381n+71qn14×37q3;q39q;q6a87qmod310.(36)

Substituting (13) and (18) into (36), we find that n=0p381n+71qn14×37q9;q96q3;q311a2q3+3qq9;q93q3;q3aq3+9q2q9;q96q3;q322×aq3+6qq9;q93q3;q38714×37q9;q9)6q3;q311a91q3+7×38qq9;q99q3;q312a90q3mod310.(37)

Congruences (9) and (10) follow from (37).

Congruence (37) also implies that n=0p3243n+152qn7×38q3;q39q;q12a90qmod310.(38)

By (13) and the binomial theorem, a3qa3q3mod9.(39)

Thanks to (38) and (39), n=0p3243n+152qn7×38q3;q39q;q12a90q3mod310.(40)

Substituting (18) into (40), we see that n=0p3243n+152qn7×38a90q3q9;q912q3;q31a2q3+3qaq3q9;q93q3;q3+9q2q9;q96q3;q3247×38q9;q912q3;q331a98q3+39qq9;q915q3;q332a97q3mod310.(41)

Congruences (11) and (12) follow from (41). This completes the proof of Theorem 1.1.

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About the article

Received: 2016-04-12

Accepted: 2016-08-22

Published Online: 2016-10-16

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0067. Export Citation

© 2016 Zhao et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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