Parity Results for Certain Partition Functions
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1 THE RAMANUJAN JOURNAL, 4, , 2000 c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Parity Results for Certain Partition Functions MICHAEL D. HIRSCHHORN School of Mathematics, UNSW, Sydney NSW, Australia 2052 m.hirschhorn@unsw.edu.au Received November 3, 1998; Accepted August 20, 1999 Abstract. Blecksmith, Brillhart and Gerst proved four congruences modulo 2 involving partition generating functions of the following sort. 1 q n S(1 n 1 2 q 3n2 (mod 2 where S :n ±(1,2,3,4or 6 (mod 12}. We give simple and uniform proofs of their congruences and of several others of the same sort. Each of these congruences yields a theorem on partitions. Thus the above congruence says that the number of partitions of n into parts not congruent to 0 or ±5 (mod 12 is odd if and only if n is a square or three times a square. Key words: restricted partitions, congruences, parity 2000 Mathematics Subject Classification: Primary 11P83 1. Introduction In an investigation of the parity of p(n, Blecksmith et al. [1] were led to discover congruences modulo 2 involving certain partition generating functions. They found that [1, Theorem 2a] 1 (1 1 2 q 2n2 (mod 2 (1 n S 1 where S 1 n > 0 : n ±(1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13 or 15 (mod 32}, [1, Theorem 2b] 1 (1 2 1 q 2n2 1 n S 2 (mod 2 (2 where S 2 n > 0 : n ±(1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 14 or 15 (mod 32}, [1, Theorem 4a] 1 (1 1 2 q 3n2 (mod 2 (3 n S 3
2 130 HIRSCHHORN where S 3 :n ±(1, 2, 3, 4 or 6 (mod 12}, and [1, Theorem 4b] 1 (1 2 1 q 3n2 1 n S 4 (mod 2 (4 where S 4 :n ±(2, 3, 4, 5 or 6 (mod 12}. We give simple and uniform proofs of these and of another six results of the same sort, namely 1 (1 1 2 q 5n2 (mod 2 (5 n S 5 where S 5 :n ±(1, 2, 8, 9 or 10 (mod 20}, 1 (1 2 1 q 5n2 1 n S 6 (mod 2 (6 where S 6 :n ±(3,4,6,7or 10 (mod 20}, 1 (1 1 2 q 2n2 q 3n2 q 6n2 (mod 2 (7 n S 7 where S 7 :n ±(1, 5, 6, 7, 8, 11, 12, 13, 17, 18, 19, 23 (mod 48}, 1 (1 2 1 q 2n2 1 q 3n2 1 q 6n2 1 n S 8 (mod 2 (8 where S 8 :n ±(1, 4, 5, 6, 7, 8, 11 or 12 (mod 24}, 1 (1 1 2 q 2n2 q 5n2 q 10n2 (mod 2 (9 n S 9 where S 9 :n ±(1, 3, 4, 7, 9, 11, 12, 13, 17, 19 or 20 (mod 40}, and 1 (1 2 1 q 2n2 1 q 5n2 1 q 10n2 1 n S 10 (mod 2 (10 where S 10 : n ±(1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19 or 20 (mod 40}. We also give a further three results of the same sort which follow from those already stated, namely 1 (1 n S 11 q (n2 n/2 (mod 2 (11
3 PARITY RESULTS FOR CERTAIN PARTITION FUNCTIONS 131 where S 11 :n ±1 or 2 (mod 4} (this is also easily proved directly, 1 (1 q 2n2 2n n S 12 q 6n2 6n1 (mod 2 (12 where S 12 :n ±(1, 2, 5, 6, 7, 10, 11 or 12 (mod 24} and 1 (1 2 n q 5n2 5n1 (mod 2 (13 n S 13 where S 13 :n ±(1, 3, 7, 9 or 10 (mod 20}. Each of (1 (13 has a corresponding interpretation in terms of partitions. Thus, for example, (3 says that The number of partitions of n into parts ±(1, 2, 3, 4 or 6 (mod 12 is odd unless n is a square or three times a square. Our main tool in proving the results (1 (10 is a pair of identities given in Lemma 1, which we prove in Section 3. Lemma 1. 1 ( aq;q 2 ( a 1 q;q 2 2 (aq;q 2 (a 1 q;q 2 ( bq; q2 ( b 1 q; q 2 } (bq; q 2 (b 1 q; q 2 (abq2 ;q 4 (a 1 b 1 q 2 ;q 4 (ab 1 q 2 ;q 4 (a 1 bq 2 ; q 4 (q 4 ; q 4 2 (aq;q 2 (a 1 q;q 2 (bq; q 2 (b 1 q; q 2 (q 2 ; q 2 2 and 1 ( aq;q 2 ( a 1 q;q 2 ( bq; q2 ( b 1 q; q 2 } 2 (aq;q 2 (a 1 q;q 2 (bq; q 2 (b 1 q; q 2 a 1 q (ab;q4 (a 1 b 1 q 4 ;q 4 (ab 1 ;q 4 (a 1 bq 4 ; q 4 (q 4 ; q 4 2. (aq;q 2 (a 1 q;q 2 (bq; q 2 (b 1 q; q 2 (q 2 ; q 2 2 We also require the fact that, modulo 2, 2 d(n 1 where d(n is the number of divisors of n, and we use without comment the fact that, modulo 2, (q; q (q; q 2 ( q; q 2 (q; q 2 (q2 ; q 2 (q; q (q; q 2 1.
4 132 HIRSCHHORN 2. Proofs of the congruences We shall prove only a few of our results. Proofs of the remaining results differ only slightly. Proof of (1: (Modulo q 2n2 1 1 q q 2n n 1 q 2n 1 n 1 n odd 1 1 ( q 8n1 q8n3 q8n5 1 q8n1 1 q8n3 1 q ( q 8n1 q8n7 2 1 q8n7 1 q 8n7 1 2 ( q 8n3 } q8n5 1 q8n3 1 q 8n 5 (1 (1 2q8n1 2q8n7 1 q 8n7 8n5 q8n q 8n1 (1 2q8n3 1 q 8n7 (1 2q8n5 } 1 q 8n3 1 q 8nb5 } 1 ( q;q 8 ( q 7 ;q 8 ( q3 ;q 8 ( q 5 ;q 8 2 (q;q 8 (q 7 ;q 8 (q 3 ;q 8 (q 5 ;q 8 (q4 ;q 16 (q 6 ; q 16 (q 10 ; q 16 (q 12 ; q 16 (q 16 ; q 16 2 (q; q 8 (q 3 ; q 8 (q 5 ; q 8 (q 7 ; q 8 (q 8 ; q 8 2 by Lemma 1 with q 4 for q, a q 3, b q (q 2 ; q 2 (q 16 ; q 16 (q; q 2 (q 2 ; q 16 (q 8 ; q 16 (q 14 ; q 16 (q 8 ; q 8 2 1/(q; q 2 (q 2 ; q 16 (q 8 ; q 16 (q 14 ; q 16 (q 2 ; q 4 } 1/(q; q 2 (q 2 ; q 16 2 (q6 ; q 16 (q 8 ; q 16 (q 10 ; q 16 (q 14 ; q 16 2 } 1/(q; q 2 (q 4 ; q 32 (q 6 ; q 16 (q 8 ; q 16 (q 10 ; q 16 (q 28 ; q 32 } 1 (1 n S 1 where S 1 :n ±(1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13 or 15 (mod 32}, as claimed.
5 PARITY RESULTS FOR CERTAIN PARTITION FUNCTIONS 133 Proof of (8: (Modulo 2 2 q 2n2 q 3n2 n 0, ±1, ±2, 3 (mod 6 q 2n q 6n2 q 3n 1 q n 1 q 2n 1 q 3n 1 q n n 0 (mod 6 n ±1 (mod n 0, ±2 (mod 6 ( q 12n1 q12n5 q12n7 1 q12n1 1 q12n5 1 q ( q 12n1 q12n q12n1 1 q 12n ( q 12n5 } q12n7 1 q12n5 1 q 12n7 (1 (1 2q12n1 2q12n11 q 6n 1 q 6n 1 q n 12n7 q12n11 n 0, ±3 (mod 6 1 q 12n q 12n1 (1 2q12n5 1 q 12n11 (1 2q12n7 } q 12n5 ( q; q 12 ( q 11 ; q 12 (q; q 12 (q 11 ; q 12 1 q 12n7 ( q5 ; q 12 ( q 7 ; q 12 (q 5 ; q 12 (q 7 ; q 12 q (q4 ; q 24 (q 6 ; q 24 (q 18 ; q 24 (q 20 ; q 24 (q 24 ; q 24 2 (q; q 12 (q 5 ; q 12 (q 7 ; q 12 (q 11 ; q 12 (q 12 ; q 12 2 q (q4 ; q 24 (q 20 ; q 24 (q 24 ; q 24 (q; q 6 (q 5 ; q 6 (q 6 ; q 6 (q 4 ; q 4 q (q; q 6 (q 5 ; q 6 (q 6 ; q 6 (q 8 ; q 24 (q 12 ; q 24 (q 16 ; q 24 (q 4 ; q 4 q (q; q 6 (q 5 ; q 6 (q 6 ; q 12 (q 8 ; q 24 (q 16 ; q 24 q/(q; q 6 (q 5 ; q 6 (q 6 ; q 12 (q 8 ; q 24 (q 16 ; q 24 (q 4 ; q 8 } q (1 n S 8 }
6 134 HIRSCHHORN where S 8 :n ±(1, 4, 5, 6, 7, 8, 11, 12 (mod 24}. Results (2 (7, (9, (10 can similarly be proved using Lemma 1 and manipulating the infinite products which arise. If we write (2 q (1 2 q 2n2 (2 n S 2 and multiply (1 and (2, we obtain 2 n odd 2 q 4n2 q (q2 ; q 2 (q 16 ; q 32 (q 32 ; q 32 (q; q 2 (q8 ; q 32 (q 24 ; q 32 q (q2 ; q 2 (q 16 ; q 16 (q; q 2 (q8 ; q 16 q/(q 8 ; q 16 (q 16 ; q 32 } q (1. n ±8,16 (mod 32 If we now divide by q and replace q 8 by q, we obtain (11. Similarly, (7 and (8 yield (12, (9 and (10 yield (13. For sake of completeness, we note that (3 and (4 yield (8, (5 and (6 yield ( Proof of Lemma 1 We shall first prove Lemma 2. ( aq;q 2 ( a 1 q;q 2 ( bq; q 2 ( b 1 q; q 2 (q 2 ; q 2 ( abq 2 ;q 4 ( a 1 b 1 q 2 ;q 4 ( ab 1 q 2 ;q 4 ( a 1 bq 2 ; q 4 (q 4 ; q 4 2 a 1 q( ab;q 4 ( a 1 b 1 q 4 ;q 4 ( ab 1 ;q 4 ( a 1 bq 4 ; q 4 (q 4 ; q 4 2. If we put b for b, then a for a in Lemma 2 and add the two results, then divide by 2(aq;q 2 (a 1 q;q 2 (bq; q 2 (b 1 q; q 2 (q 2 ; q 2 2, we obtain the first identity of Lemma 1. If instead we subtract, we obtain the second identity of Lemma 1.
7 PARITY RESULTS FOR CERTAIN PARTITION FUNCTIONS 135 Proof of Lemma 2: We have ( a 2 q 2 ; q 4 ( a 2 q 2 ; q 4 ( b 2 q 2 ; q 4 ( b 2 q 2 ; q 4 (q 4 ; q 4 2 a 2r b 2s q 2r 2 2s 2 r,s r,s u,v u v (mod 2 u,v even m,n a 2r b 2s q (rs2 (r s 2 a uv b u v q u2 v 2 (ab u q u2 (ab 1 v q v2 (ab u q u2 (ab 1 v q v2 u,v odd (ab 2m q 4m2 (ab 1 2n q 4n2 m,n (ab 2m 1 q 4m2 4m1 (ab 1 2n 1 q 4n2 4n1 ( a 2 b 2 q 4 ;q 8 ( a 2 b 2 q 4 ;q 8 ( a 2 b 2 q 4 ;q 8 ( a 2 b 2 q 4 ;q 8 (q 8 ;q 8 2 a 2 q 2 ( a 2 b 2 ;q 8 ( a 2 b 2 q 8 ;q 8 ( a 2 b 2 ;q 8 ( a 2 b 2 q 8 ;q 8 (q 8 ;q 8 2. Now put a for a 2, b for b 2 and q for q 2. Reference 1. R. Blecksmith, J. Brillhart, and I. Gerst, Parity results for certain partition functions and identities similar to theta function identities, Mathematics of Computation, 48(177 (1987,
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