CONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS

Bull. Aust. Math. Soc. 9 2016, 400 409 doi:10.1017/s000497271500167 CONGRUENCES MODULO 2 FOR CERTAIN PARTITION FUNCTIONS M. S. MAHADEVA NAIKA, B. HEMANTHKUMAR H. S. SUMANTH BHARADWAJ Received 9 August 2015; accepted 5 September 2015; first published online 11 November 2015 Abstract Let b,5 n denote the number of partitions of n into parts that are not multiples of or 5. We establish several infinite families of congruences modulo 2 for b,5 n. In the process, we also prove numerous parity results for broken 7-diamond partitions. 2010 Mathematics subject classification: primary 11P8; secondary 05A17. Keywords phrases: congruence, partition, theta function. 1. Introduction For any positive integer l 2, a partition is said to be l-regular if none of its parts is a multiple of l. The number of l-regular partitions of n is denoted by b l n by convention we define b l 0 = 1. The generating function of b l n satisfies the identity b l nq n = f l. f 1 Here throughout this paper, we use the notation f k := q k ; q k k = 1, 2,..., where a; q = 1 aq m. Let f a, b be Ramanujan s general theta function [4, page 4] given by f a, b := a nn+1/2 b nn 1/2. n= Jacobi s triple product identity states that f a, b = a; ab b; ab ab; ab. The first author was supported by DST, Government of India No. SR/S4/MS:79/11 the third author by a CSIR Junior Research Fellowship No. 09/090111/2014-EMR-I. c 2015 Australian Mathematical Publishing Association Inc. 0004-9727/2015 $16.00 m=0 400

[2] Congruences modulo 2 for certain partition functions 401 In particular, f q := f q, q 2 = 1 n q nn 1/2 = q; q. n= Ramanujan [1, page 212] stated the following identity without proof: f 1 = f 25 Rq q q 2 R 1 q, where Rq = f q15, q 10 f q 20, q 5. 1.1 Watson [14] gave a proof of 1.1 utilising the quintuple product identity q n2 +n/2 z n z n 1 = q; q zq; q 1/z; q z 2 q; q 2 q/z 2 ; q 2. n= For any prime p 5, Cui Gu [6] obtained the p-dissection formula f q = p 1/2 k= p 1/2 k ±p 1/6 1 k q k2 +k/2 f q p2 +6k+1p/2, q p2 6k+1p/2 + 1 ±p 1/6 q p2 1/24 f q p2, 1.2 where the choice of the ± sign is made so that ±p 1/6 is an integer. Note that k 2 + k/2 p 2 1/24 mod p as k runs through the range of the summation. Recently, numerous infinite families of congruences modulo 2 for b l n have been established. For example, Xia Yao [15] proved that b 9 2 6 j+4 n + 5 26 j+ 1 0 mod 2 for all nonnegative integers j n, while Cui Gu [6] showed that b 5 4 5 2 j+1 n + 1 52 j 1 0 mod 2. 6 For more examples, see [5, 7 9]. MacMahon s partition analysis guided Andrews Paule [2] to introduce broken k-diamond partitions. For any positive integer k, let k n denote the number of broken k-diamond partitions of n. The generating function of k n satisfies the identity k nq n = f 2 f 2k+1 f 1 f. 1. 4k+2 Various authors have obtained parity results for broken k-diamond partitions see, for example, [10, 12, 16]. In particular, Ahmed Baruah [1] established 7 8n + 2 7 64n + 54 0 mod 2 1.4

402 M. S. Mahadeva Naika et al. [] 7 8 5 2 j+1 n + 24r + 16 52 j + 2 0 mod 2 for all j, n 0 r {, 4}. With this motivation, we obtain several infinite families of congruences for the broken 7-diamond partitions for the partition function b,5 n which counts the number of partitions of n none of whose parts are multiples of or 5. For example, we have the following results. Theorem 1.1. If p is an odd prime with 15/p = 1, 1 i p 1 j, n 0, b,5 2 p 2 j+2 n + 6i + 2p p2 j+1 + 1 0 mod 2. 1.5 Theorem 1.2. For all integers n, j 0, 7 2 6 j+ n + 26 j+2 + 2 0 mod 2 1.6 7 2 6 j+6 n + 5 26 j+5 + 2 0 mod 2. 1.7 Note that congruences 1.4 are special cases of 1.6 1.7. We prove Theorem 1.1 more families of congruences for b,5 n in Section Theorem 1.2 further infinite families of congruences for broken 7-diamond partitions in Section 4. 2. Preliminaries In this section we present two lemmas which play a vital role in proving our main results. Note that the generating function for b,5 n satisfies Lemma 2.1. We have where dn is defined by f f 5 f 1 f 15 f 1 f 15 f f 5 b,5 nq n = f f 5 f 1 f 15. 2.1 b,5 2nq 2n + q f 6 f 10 mod 2 2.2 d2nq 2n q f 2 f 0 mod 2, 2. dnq n = f 1 f 15 f f 5.

[4] Congruences modulo 2 for certain partition functions 40 Proof. From 2.1, b,5 nq n = f f 5 = f 6 f 10 q ; q 6 q 5 ; q 10, f 1 f 15 f 2 f 0 q; q 2 q 15 ; q 0 which implies that 2 b,5 2n + 1q 2n+1 = f 2 6 f 10 2 f 4 f { 60 q; q 2 q 15 ; q 0 f2 2 f 0 2 f q; q2 q 15 ; q 0 }. 12 f 20 q ; q 6 q 5 ; q 10 q ; q 6 q 5 ; q 10 2.4 Using the modular equation of degree 15 [4, Entry 11iv, page 8], Baruah Berndt [] derived the identity { q; q 2 q 15 ; q 0 q; q2 q 15 ; q 0 } = 2q f 12 f 20. 2.5 q ; q 6 q 5 ; q 10 q ; q 6 q 5 ; q 10 f 6 f 10 In view of 2.4 2.5, we can easily see that b,5 2n + 1q 2n+1 = q f 4 f 6 f 10 f 60 f2 2 f. 2.6 0 2 From 2.1 2.6, f f 5 f 1 f 15 = b,5 2nq 2n + q f 4 f 6 f 10 f 60 f2 2 f. 2.7 0 2 By the binomial theorem, we can see that for all positive integers k m, f 2m k f m 2k mod 2. 2.8 Congruence 2.2 follows from 2.7 2.8. Replacing q by q in 2.7 using the relation f 1 f 15 f f 5 = q; q = f 2 f 1 f 4, Congruence 2. readily follows from 2.9 2.8. d2nq 2n q f 2 f 0 f 12 f 20 f 2 6 f. 2.9 10 2 Lemma 2.2. Set p [l 1 m 1 ]nq n = f l f m. Then f f 5 = p [ 1 5 1 ]2nq 2n q f 12 f 20 + q f 2 f 0 2.10 f 1 f 15 = f 6 f 10 q 2 f 4 f 60 + p [1 1 15 1 ]2n + 1q 2n+1. 2.11

404 M. S. Mahadeva Naika et al. [5] Proof. In [11], we used Ramanujan s theta function identities to derive the identities q ; q 6 q 5 ; q 10 q ; q 6 q 5 ; q 10 = 2q f 12 f 20 f 6 f 10 2q f 2 f 0 f 6 f 10 2.12 q; q 2 q 15 ; q 0 + q; q 2 q 15 ; q 0 = 2 f 6 f 10 f 2 f 0 2q 2 f 4 f 60 f 2 f 0. 2.1 Equations 2.10 2.11 follow easily from 2.12 2.1, respectively.. Congruences for b,5 n modulo 2 In this section, we prove infinite families of congruences modulo 2 for b,5 n. Theorem.1. Let j 0. Then b,5 2 5 2 j n + 2 52 j + 1 q n f f 5 mod 2.1, for all n 0, r {14, 26} s {22, 28}, b,5 2 5 2 j+1 n + r 52 j + 1 0 mod 2.2 b,5 2 5 2 j+2 n + s 52 j+1 + 1 0 mod 2.. Proof. From the equations 2.1 2.2, b,5 2n + 1 q n f f 5 mod 2,.4 which is the j = 0 case of.1. Suppose that the congruence.1 holds for some integer j 0. Using 1.1, b,5 2 5 2 j n + 2 52 j + 1 q n f 5 f 75 Rq q q 6 R 1 q mod 2..5 Extracting the terms in q n with n mod 5 using 1.1 again yields b,5 2 5 2 j 5n + + 2 52 j + 1 q n = b,5 2 5 2 j+1 n + 4 52 j+1 + 1 q n f 1 f 15 f 15 f 25 Rq q q 2 R 1 q mod 2..6

[6] Congruences modulo 2 for certain partition functions 405 In the same way, using.6, b,5 2 5 2 j+1 5n + 1 + 4 52 j+1 + 1 = b,5 2 5 2 j+2 n + 2 52 j+2 + 1 q n q n f f 5 mod 2. Thus,.1 is true for j + 1. Hence, by induction,.1 is true for any nonnegative integer j. Congruences.2. follow from.5.6, respectively. Theorem.2. For any prime p 5 with 15/p = 1 any integers j, n 0, b,5 2 p 2 j n + 2 p2 j + 1 q n f f 5 mod 2..7 Proof. Note that.4 is the j = 0 case of.7. Consider the congruence k2 + k 2 + 5 m2 + m 2 p2 1 where p 1/2 k, m p 1/2. We can rewrite.8 as 18k + 2 + 156m + 1 2 0 mod p. mod p,.8 Since 15/p = 1, the only solution of this congruence is k = m = ±p 1/6. This fact along with 1.2.4 yields b,5 2 p 2 n + p2 1 + 1 q n f f 5 mod 2, which is the j = 1 case of.7. Iterating this procedure yields Theorem.2. Proof of Theorem 1.1. From the proof of Theorem.2 it is clear that the congruence.8 holds only when k = m = ±p 1/6. Thus, employing 1.2 in.7, b,5 2 p 2 j = pn + p2 1 + 2 p2 j + 1 q n b,5 2 p 2 j+1 n + 2 p2 j+2 + 1 q n f p f 5p mod 2, which implies that, for i = 1, 2,..., p 1, b,5 2 p 2 j+1 pn + i + 2 p2 j+2 + 1 0 mod 2, from which the congruence 1.5 follows immediately.

406 M. S. Mahadeva Naika et al. [7] Theorem.. For all integers n 0 j 0, b,5 2 6 j+1 n + 26 j+1 + 1 b,5 2n + 1 mod 2,.9 b,5 2 6 j+4 n + 26 j+ + 1 0 mod 2,.10 b,5 2 6 j+7 n + 5 26 j+6 + 1 0 mod 2,.11 b,5 2 6 j+6 n + 26 j+5 + 1 b,5 2 6 j+2 n + 26 j+1 + 1 mod 2,.12 b,5 2 6 j+5 n + 5 26 j+4 + 1 b,5 2 6 j+ n + 5 26 j+2 + 1 mod 2..1 Proof. By using 2.10 in.4 extracting the terms involving odd powers of q, which yields b,5 4n + q 2n+1 q f 2 f 0 q f 12 f 20 mod 2, b,5 4n + q n f 1 f 15 f 6 f 10 mod 2..14 By substituting 2.11 in.14 extracting the terms involving even powers of q, b,5 8n + q n q f 2 f 0 mod 2..15 It follows from.15 that b,5 16n + 11q n f 1 f 15 mod 2.16 In view of.14.16, b,5 16n + 0 mod 2..17 b,5 2n + 27 b,5 8n + 7 mod 2..18 Using 2.11 in.16 extracting the terms involving even powers of q, b,5 2n + 11q n f f 5 q f 2 f 0 mod 2..19 It follows from.4.19 that b,5 64n + 11 b,5 4n + 1 mod 2..20

[8] Congruences modulo 2 for certain partition functions 407 By substituting 2.10 in.19 extracting the terms containing odd powers of q, b,5 64n + 4q n f 6 f 10 mod 2..21 From.21, for all n 0 By.4.2, b,5 128n + 107 0 mod 2.22 b,5 128n + 4q n f f 5 mod 2..2 b,5 128n + 4 b,5 2n + 1 mod 2..24 Congruence.9 follows from.24 mathematical induction. Using.17.22 in.9, we deduce.10.11, respectively. Congruence.12 follows from.9.20. Congruence.1 follows from.9.18. 4. Parity results for broken 7-diamond partitions In this section, we prove infinite families of congruences modulo 2 for 7 n, using our results for b,5 n. Theorem 4.1. For any integers n 0 j 0, 7 2 6 j+2 n + 26 j+2 + 2 b,5 8n + mod 2. 4.1 Proof. Setting k = 7 in 1., 7 nq n = f 2 f 15 f 1 f 1 mod 2, 0 f 1 f 15 which implies that 7 2nq 2n 1 { q; q 2 q 15 ; q 0 + q; q 2 q 15 ; q 0 } mod 2. 4.2 2 f 2 f 0 q 2 ; q 4 q 0 ; q 60 Using equation 2.1 in 4.2 replacing q 2 by q, 7 2nq n f2 f f 5 f 0 f 1 f q f 2 2 f 2 0 f f 5 15 f 1 f q f 1 f 15 mod 2. 4. f 15 1 f 15 Using equation 2.2 2.11 in 4., 7 4n + 2q 2n+1 q f 4 f 60 mod 2. 4.4 From.15 4.4, Congruence 4.1 follows from.9 4.5. 7 4n + 2 b,5 8n + mod 2. 4.5

408 M. S. Mahadeva Naika et al. [9] Proof of Theorem 1.2. From 4.1, 7 2 6 j+2 2n + 26 j+2 + 2 b,5 82n + mod 2 4.6 7 2 6 j+2 16n + 1 + 26 j+2 + 2 b,5 816n + 1 + mod 2. 4.7 Congruence 1.6 follows readily from.17 4.6. Congruence 1.7 follows from.22 4.7. Theorem 4.2. For any prime p 5 with 15/p = 1 any integers j, n 0, 7 4 p 2 j n + 4 p2 j + 2 b,5 8n + mod 2. 4.8 Proof. Combining equations.4.7 with 4.5 yields 4.8. Theorem 4.. If p is an odd prime with 15/p = 1 1 i p 1, then, for all j, n 0, 7 8 p 2 j+2 n + 24i + 16p p2 j+1 + 2 0 mod 2. Proof. We omit the proof, since it is similar to the proof of Theorem 1.1. Acknowledgement The authors would like to thank the anonymous referee for his/her valuable comments suggestions. References [1] Z. Ahmed N. D. Baruah, Parity results for broken 5-diamond, 7-diamond 11-diamond partitions, Int. J. Number Theory 11 2015, 527 542. [2] G. E. Andrews P. Paule, MacMahon s partition analysis XI: broken diamonds modular forms, Acta Arith. 126 2007, 281 294. [] N. D. Baruah B. C. Berndt, Partition identities Ramanujan s modular equations, J. Combin. Theory Ser. A 114 2007, 1024 1045. [4] B. C. Berndt, Ramanujan s Notebooks, Part III Springer, New York, 1991. [5] N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston J. Radder, Divisibility properties of the 5-regular 1-regular partition functions, Integers 8 2008, A60. [6] S. P. Cui N. S. S. Gu, Arithmetic properties of l-regular partitions, Adv. Appl. Math. 51 201, 507 52. [7] D. Furcy D. Penniston, Congruences for l-regular partition functions modulo, Ramanujan J. 27 2012, 101 108. [8] M. D. Hirschhorn J. A. Sellers, Elementary proofs of parity results for 5-regular partitions, Bull. Aust. Math. Soc. 81 2010, 58 6. [9] W. J. Keith, Congruences for 9-regular partitions modulo, Ramanujan J. 5 2014, 157 164. [10] B. L. S. Lin, Elementary proofs of parity results for broken -diamond partitions, J. Number Theory 15 2014, 1 7.

[10] Congruences modulo 2 for certain partition functions 409 [11] M. S. Mahadeva Naika, B. Hemanthkumar H. S. Sumanth Bharadwaj, Color partition identities arising from Ramanujan s theta functions, Acta Math. Vietnam., to appear. [12] S. Radu J. A. Sellers, Parity results for broken k-diamond partitions 2k + 1-cores, Acta Arith. 146 2011, 4 52. [1] S. Ramanujan, Collected Papers Cambridge University Press, Cambridge, 1927. [14] G. N. Watson, Ramanujan s Vermutung über Zerfällungsanzahlem, J. reine angew. Math. 179 198, 97 128. [15] E. X. W. Xia O. X. M. Yao, Parity results for 9-regular partitions, Ramanujan J. 4 2014, 109 117. [16] O. X. M. Yao, New parity results for broken 11-diamond partitions, J. Number Theory 140 2014, 267 276. M. S. MAHADEVA NAIKA, Department of Mathematics, Bangalore University, Central College Campus, Bengaluru-560 001, Karnataka, India e-mail: msmnaika@rediffmail.com B. HEMANTHKUMAR, Department of Mathematics, Bangalore University, Central College Campus, Bengaluru-560 001, Karnataka, India e-mail: hemanthkumarb.0@gmail.com H. S. SUMANTH BHARADWAJ, Department of Mathematics, Bangalore University, Central College Campus, Bengaluru-560 001, Karnataka, India e-mail: sumanthbharadwaj@gmail.com