Some congruences for 3-component multipartitions

Tao Yan Zhao; Lily J. Jin; C. Gu

doi:10.1515/math-2016-0067

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Some congruences for 3-component multipartitions

Tao Yan Zhao , Lily J. Jin and C. Gu

Published/Copyright: October 16, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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,…, λ_k) of integers λ_i (i = 1,2,…, k) is a partition of the natural number n if n=∑i=1kλi and λ₁ ≥ λ₂ ≥ ··· ≥ λ_k > 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 p_r(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 p_r(0) = 1. The generating function of p_r(n) is

∑n=0∞prnqn=1q;q∞r,(1)

where

(a;q)∞=∏n=1∞1−aqn.(2)

A number of congruences satisfied by p_r(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+5≡0mod33(3)

and

p39n+8≡0mod34.(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+17≡0mod35,(5)

p327n+26≡0mod37,(6)

p381n+44≡0mod36,(7)

p381n+71≡0mod37,(8)

p3243n+152≡0mod38,(9)

p3243n+233≡0mod310,(10)

p3729n+395≡0mod39,(11)

p3729n+638≡0mod310.(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;q9∞3q3;q3∞(13)

and

bq=aq3−3qq9;q9∞3q3;q3∞(14)

where a(q) and b(q) are defined by

aq=∑m,n=−∞∞qm2+mn+n2(15)

and

bq=∑m,n=−∞∞ωm−nqm2+mn+n2,ω=exp2πi/3.(16)

Lemma 2.1 was proved by Borwein, Borwein and Garvan [18].

Lemma 2.2

We have

bq=q;q∞3q3;q3∞.(17)

Lemma 2.2 was also proved by Borwein, Borwein and Garvan [18].

Lemma 2.3

The following 3-dissection formula holds:

1q;q∞3=q9;q9∞3q3;q3∞10a2q3+3qaq3q9;q9∞3q3;q3∞+9q2q9;q9∞6q3;q3∞2.(18)

Proof

It is easy to check that for any positive integer n,

1−qn1−ωqn1−ω2qn=1−qn3,if3|n1−q3n,if3∤n.(19)

By (17) and (19),

bqbωqbω2q=q;q∞3ωq;ωq∞3ω2q;ω2q∞3q3;q3∞3=1q3;q3∞3∏n=1∞1−qn1−ωqn1−ω2qn3=1q3;q3∞3∏3∤n1−q3n∏3∤n1−qn33=1q3;q3∞3q3;q9∞q6;q9∞q3;q3∞33=q3;q3∞9q9;q9∞3.(20)

In view of (14), we find that

bωqbω2q=aq3−3ωqq9;q9∞3q3;q3∞aq3−3ω2qq9;q9∞3q3;q3∞=a2q3+3qq9;q9∞3q3;q3∞aq3+9q2q9;q9∞6q3;q3∞2.(21)

It follows from (17), (20) and (21) that

1q;q∞3=1q3;q3∞bq=bωqb(ω2q)q3;q3∞bqbωqbω2q=q9;q9∞3q3;q3∞10a2q3+3qaq3q9;q9∞3q3;q3∞+9q2q9;q9∞6q3;q3∞2.(22)

Remark

Baruah and Ojah [7] also deduced the following 3-dissection formula for 1q;q∞3, 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=0∞p3(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=0∞p3(3n+2)qn=9q3;q3∞9q;q∞12.(26)

Substituting (18) into (26), we deduce that

∑n=0∞p33n+2qn=9q9;q9∞12q3;q3∞31a2q3+3qaq3q9;q9∞3q3;q3∞+9q2(q9;q9)∞6(q3;q3)∞24≡9q9;q9∞12q3;q3∞31a8q3+108qq9;q9∞15q3;q3∞32a7q3+810q2q9;q9∞18q3;q3∞33a6q3+3888q3q9;q9∞21q3;q3∞34a5q3+13851q4q9;q9∞24q3;q3∞35a4q3+34992q5q9;q9∞27q3;q3∞36a3q3+6561q6q9;q9∞30q3;q3∞37a2q3+19683q7q9;q9∞33q3;q3∞38aq3mod310,(27)

which yields

∑n=0∞p39n+8qn≡810q3;q3∞18q;q∞33a6q+34992qq3;q3∞27q;q∞36a3qmod310.(28)

Substituting (13) and (18) into (28), we see that

∑n=0∞p39n+8qn≡810q9;q9∞33q3;q3∞92a2q3+3qq9;q9∞3q3;q3∞aq3+9q2q9;q9∞6q3;q3∞211×aq3+6qq9;q9∞3q3;q3∞6+34992qq9;q9∞36q3;q3∞93a2q3+3qq9;q9∞3q3;q3∞aq3+9q2q9;q9∞6q3;q3∞212×aq3+6qq9;q9∞3q3;q3∞3≡810q9;q9∞33q3;q3∞92a28q3+31833qq9;q9∞36q3;q3∞93a27q3+50301q2q9;q9∞39q3;q3∞94a26q3+52488q3q9;q9∞42q3;q3∞95a25q3+39366q4q9;q9∞45q3;q3∞96a24q3mod310,(29)

which implies that

∑n=0∞p327n+17qn≡131×35q3;q3∞36q;q∞93a27q+2×39qq3;q3∞45q;q∞96a24qmod310(30)

and

∑n=0∞p327n+26qn≡23×37q3;q3∞39q;q∞94a26qmod310.(31)

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

Substituting (13) and (18) into (30), we get

∑n=0∞p327n+17qn≡131×35q9;q9∞93q3;q3∞274a2q3+3qq9;q9∞3q3;q3∞aq3+9q2q9;q9∞6q3;q3∞231×aq3+6qq9;q9∞3q3;q3∞27+2×39qq9;q9∞96q3;q3∞275a2q3+3qq9;q9∞3q3;q3∞aq3+9q2q9;q9∞6q3;q3∞232×aq3+6qq9;q9∞3q3;q3∞24≡31833q9;q9∞93q3;q3∞274a89q3+8019qq9;q9∞96q3;q3∞275a88q3+30618q2q9;q9∞99q3;q3∞276a87q3+52488q3q9;q9∞102q3;q3∞277a86q3+39366q4q9;q9∞105q3;q3∞278a85q3mod310.(32)

It follows from (32) that

∑n=0∞p381n+44qn≡11×36q3;q3∞96q;q∞275a88q+2×39qq3;q3∞105q;q∞278a85qmod310(33)

and

∑n=0∞p381n+71qn≡14×37q3;q3∞99q;q∞276a87qmod310.(34)

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

By the binomial theorem,

q3;q3∞90q;q∞270≡1mod27.(35)

Combining (34) and (35), we have

∑n=0∞p381n+71qn≡14×37q3;q3∞9q;q∞6a87qmod310.(36)

Substituting (13) and (18) into (36), we find that

∑n=0∞p381n+71qn≡14×37q9;q9∞6q3;q3∞11a2q3+3qq9;q9∞3q3;q3∞aq3+9q2q9;q9∞6q3;q3∞22×aq3+6qq9;q9∞3q3;q3∞87≡14×37q9;q9)∞6q3;q3∞11a91q3+7×38qq9;q9∞9q3;q3∞12a90q3mod310.(37)

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

Congruence (37) also implies that

∑n=0∞p3243n+152qn≡7×38q3;q3∞9q;q∞12a90qmod310.(38)

By (13) and the binomial theorem,

a3q≡a3q3mod9.(39)

Thanks to (38) and (39),

∑n=0∞p3243n+152qn≡7×38q3;q3∞9q;q∞12a90q3mod310.(40)

Substituting (18) into (40), we see that

∑n=0∞p3243n+152qn≡7×38a90q3q9;q9∞12q3;q∞31a2q3+3qaq3q9;q9∞3q3;q3∞+9q2q9;q9∞6q3;q3∞24≡7×38q9;q9∞12q3;q3∞31a98q3+39qq9;q9∞15q3;q3∞32a97q3mod310.(41)

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

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Received: 2016-4-12

Accepted: 2016-8-22

Published Online: 2016-10-16

Published in Print: 2016-1-1

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