PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND (2k + 1) CORES

Transcription

1 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES SILVIU RADU AND JAMES A. SELLERS Abstract. In this paper we prove several new parity results for broken k-diamond partitions introduced in 2007 by Andrews and Paule. In the process, we also prove numerous congruence properties for 2k+-core partitions. The proof technique involves a general lemma on congruences which is based on modular forms.. Introduction Broken k-diamond partitions were introduced recently by Andrews and Paule []. These are constructed in such a way that the generating functions of their counting sequences k n n 0 are closely related to modular forms. Namely, k nq n = where we recall the Dedekind eta function ητ := q 24 q 2n q 2k+n q n 3 q 4k+2n k+/2 η2τη2k + τ = q ητ 3 η4k + 2τ, k, q n q = e 2πiτ. In [], Andrews and Paule proved that, for all n 0, 2n + 0 mod 3 and conjectured a few other congruences modulo 2 satisfied by certain families of k-broken diamond partitions. Since then, a number of authors have provided proofs of additional congruences satisfied by broken k-diamond partitions. Hirschhorn and Sellers [8] provided a new proof of the modulo 3 result mentioned above as well as elementary proofs of the following parity results: For all n, 4n mod 2, 4n mod 2, 2 0n mod 2, 2 0n mod 2 The third result in the list above appeared in [] as a conjecture while the other three did not. Soon after the publication of [8], Chan [3] provided a different proof of the parity results for 2 mentioned above as well as a number of congruences modulo powers of 5. In this paper, we significantly extend the list of known parity results for broken k-diamonds by proving a large number of congruences which are similar to those mentioned above. Indeed, we will do so by proving a similar set of parity results satisfied by certain t-core partitions. A partition is called a t-core if none of its hook lengths is divisible by t. These partitions have been studied extensively by many, especially thanks to their strong connection to representation theory. Numerous congruence properties are known for t-cores, although few such results are known modulo S. Radu was supported by DK grant W24-DK6 of the Austrian Science Funds FWF.

2 2 SILVIU RADU AND JAMES A. SELLERS 2. Such parity results can be found in [7], [5], [9], [6], [2], [4]. In all of these papers, the value of t which was considered was even; in this paper, we provide a new set of parity results for t-cores wherein t is odd. The generating function for t-core partitions for a fixed t is given by a t nq n q tn t = q n. Given this fact, we can quickly see a connection between broken k-diamonds and 2k + -cores which we will utilize below. Lemma.. For all k we have q 4k+2n k+ k nq n a 2k+ nq n mod 2. Proof. Using the relation q n 2 q 2n mod 2 we find q 4k+2n k+ k nq n q 4k+2n k q 2n q 2k+n = q n 3 q 2k+n 2k+ q n mod 2 = a 2k+ nq n. We assume throughout that k v = a k v = 0 if v 0. Corollary.2. Let r N. Then for all k and n Z we have k 4k + 2n + r 0 mod 2 a 2k+ 4k + 2n + r 0 mod 2. Proof. Let k and r be fixed and assume that k 4k + 2n + r 0 mod 2 for all n Z. Let bnq 4k+2n = q 4k+2n k+. n Z Then using Lemma. we find that a 2k+ 4k + 2n + rq n n Z n, m Z, 4k + 2n + m r mod 4k + 2 bn k mq 4k+2n+m n, m Z, m r mod 4k + 2 bn k 4k + 2v + rq 4k+2m+4k+2v+r n,v Z 0 mod 2. bn k mq 4k+2n+m

3 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES 3 The reverse direction is analogous. With this motivation, we now state the full list of parity results we will prove in this paper. With the goal of minimizing the notation, we will write to mean that, for each i {, 2,..., m}, Theorem.3. For all n 0, ftn + r, r 2,..., r m 0 mod 2 2 0n + 2, 6 0 mod 2, ftn + r i 0 mod n + 7, 9, 3 0 mod 2, 5 22n + 2, 8, 2, 4, 6 0 mod 2, 6 26n + 2, 0, 6, 8, 20, 22 0 mod 2, 8 34n +, 5, 7, 9, 25, 27, 29, 33 0 mod 2, 9 38n + 2, 8, 0, 20, 24, 28, 30, 32, 34 0 mod 2, 46n +, 5, 2, 23, 29, 3, 35, 39, 4, 43, 45 0 mod 2. Note that was proved in [8]. Thanks to Corollary.2, we see that Theorem.3 is proved once we prove the following corresponding theorem involving t-cores: Theorem.4. For all n 0, a 5 0n + 2, 6 0 mod 2, a 7 4n + 7, 9, 3 0 mod 8, a 22n + 2, 8, 2, 4, 6 0 mod 2, a 3 26n + 2, 0, 6, 8, 20, 22 0 mod 2, a 7 34n +, 5, 7, 9, 25, 27, 29, 33 0 mod 8, a 9 38n + 2, 8, 0, 20, 24, 28, 30, 32, 34 0 mod 2, a 23 46n +, 5, 2, 23, 29, 3, 35, 39, 4, 43, 45 0 mod 8. Note that every prime p, 5 p 23, is represented in Theorem.4, which helps to explain why certain families of broken k diamond partitions appear in Theorem.3 and others do not. Our ultimate goal now is to provide a proof of Theorem.4. We close this section by developing the machinery necessary to prove this theorem. For M a positive integer let RM be the set of integer sequences indexed by the positive divisors δ of M. Let = δ,..., δ k = M be the positive divisors of M and r RM. Then we will write r = r δ,..., r δk. For s an integer and m a positive integer we denote by [s] m the set of all elements congruent to s modulo m, in other words [s] m Z m. Let Z m be the set of all invertible elements in Z m. Let S m Z m be the set of all squares in Z m. Definition.5. For m, M N,r δ RM and t {0,..., m } we define the map : S 24m {0,..., m } {0,..., m } with [s] 24m, t [s] 24m t and the image is uniquely determined by the relation [s] 24m t ts + s 24 δ M δr δ mod m. We define the set P m,r t := {[s] 24m t [s] 24m S 24m }. Lemma.6. Let p 5 be a prime. Let r p := r p, rp p =, p Rp. Then { 24t 24t 5 P 2p,r pt = t } =, t t mod 2, 0 t 2p. p p

4 4 SILVIU RADU AND JAMES A. SELLERS Proof. First note that 24 δ p δr p δ = p2 Z. 24 Let m = 2p. If s s 2 mod m then [s ] 24m t = [s 2 ] 24m t because p2 24 is an integer. This implies that 6 P 2p,r pt = {t t ts + s p2 24 We see that 7 P 2p,r pt mod 2 = {t mod 2}. mod p, s S m, 0 t 2p }. Next we compute P 2p,r pt mod p. By 6 we know { P 2p,r pt mod p = t mod p t ts + s p mod p, s S p } = {t mod p 24t s24t mod p, s S p } { 24t 24t = t } mod p =. p p By 7 and 8 and the Chinese remainder theorem we obtain P 2p,r pt mod 2p and we obtain the formula 5 by imposing that the elements of P 2p,r pt lie between 0 and 2p. We now use Lemma.6 to compute P 2p,r pt for p = 5, 7,, 3, 9, 23 and t = 2, 7, 2, 2,, 2, below. 24t p = 5, t = 2. We see that p = 2 5 =. For t 24t {, 2} we have 5 = and for t {0, 3, 4} we have {0, }. This implies that P 0,r 52 {, 2} mod 5. Since t 0 24t 5 mod 2 we have that P 0,r 52 0 mod 2. Hence by Lemma.6 we have p = 7, t = 7. We see that 24t p P 0,r 52 = {2, + 5} = {2, 6}. = 7 =. We see that for t 24t {0, 2, 6} we have 7 = {0, 2, 6} mod 7. Because t mod 2 we obtain and this is all t with this property so P 4,r 7 by Lemma.6 P 4,r 77 = {0 + 7, 2 + 7, 6 + 7} = {7, 9, 3}. p =, t = 2. Here 24t Similarly we get by Lemma.6 and = 5 2 =. We see that for t {, 2, 3, 5, 8} we have P 22,r 2 = { +, 2, 3 +, 5 +, 8} = {2, 8, 2, 4, 6}. P 26,r 3 = {2, 0, 6, 8, 20, 22}, P 34,r 7 = {, 5, 7, 9, 25, 27, 29, 33}, P 38,r 9 = {2, 8, 0, 20, 24, 28, 30, 32, 34} P 46,r 23 = {, 5, 2, 23, 29, 3, 35, 39, 4, 43, 45}. 24t = so We see immediately from the above that Theorem.4 is equivalent to the following theorem.

5 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES 5 Theorem.7. Let t : {5, 7,, 3, 7, 9, 23} {2, 7, } with p t p be defined by t 5, t 7, t, t 3, t 7, t 9, t 23 := 2, 7, 2, 2,, 2,. Then for all n 0, p prime with 5 p 23, and t P 2p,r pt p, we have 9 a p 2pn + t 0 mod 2 ip, where ip = { if p = 5,, 3, 9, 3 if p = 7, 7, 23. For each r RM we assign a generating function f r q := q δn r δ = c r nq n. δ M Given p a prime, m N and t {0,..., m } we are concerned with proving congruences of the type c r mn + t 0 mod p, n N. The congruences we are concerned with here have some additional structure; namely a r mn + t 0 mod p, n 0, t P m,r t. In other words a congruence is a tuple r, M, m, t, p with r RM, m, t {0,..., m } and p a prime such that a r mn + t 0 mod p, n 0, t P t. Throughout when we say that a r mn + t 0 mod p we mean that a r mn + t 0 mod p for all n 0 and all t P t. The purpose of this paper is show the congruences a p 2pn + t p 0 mod 2 when p = 5, 7,, 3, 7, 9, 23 and t p = 2, 7, 2, 2,, 2,. In order to accomplish our goal we need a lemma [0, Lemma 4.5]. We first state it and then explain the terminology. Lemma.8. Let u be a positive integer, m, M, N, t, r = r δ, a = a δ RN, n the number of double cosets in Γ 0 N\Γ/Γ and {γ,..., γ n } Γ a complete set of representatives of the double coset Γ 0 N\Γ/Γ. Assume that p m,r γ i + p aγ i 0, i {,..., n}. Let t min := min t P m,rt t and Then if ν := 24 for all t P m,r t then for all t P m,r t. a δ + δ N δ M ν r δ [Γ : Γ 0 N] δ N δa δ c r mn + t q n 0 mod u c r mn + t q n 0 mod u 24m δ M δr δ t min m. The lemma reduces the proof of a congruence modulo u to checking that finitely many values are divisible by u. We first define the set. Let κ = κm = gcdm 2, 24 and πm, r δ := s, j where s is a non-negative integer and j an odd integer uniquely determined by δ M δ r δ = 2 s j. Then a tuple m, M, N, r δ, t belongs to iff m, M, N, r δ RM, t {0,..., m }; p m implies p N for every prime p; δ M implies δ mn for every δ such that r δ 0; κn δ M r δ mn δ 0 mod 24;

6 6 SILVIU RADU AND JAMES A. SELLERS κn δ M r δ 0 mod 8; 24m gcdκ 24t δ M δr δ, 24m N; for s, j = πm, r δ we have 4 κn and 8 Ns or 2 s and 8 N j. Next we need to define the groups Γ, Γ 0 N and Γ : { } a b Γ := a, b, c, d Z, ad bc =, c d { } a b Γ 0 N := Γ N c c d for N a positive integer, and { } h Γ := h Z. 0 For the index we have [Γ : Γ 0 N] := N p N + p see, for example, []. a b Finally for m, M, and r RM and γ = we define c d 20 p m,r γ := min λ {0,...,m } 24 and p rγ := 24 δ M δ M r δ gcd 2 δa + κλc, mc δm r δ gcd 2 δ, c. δ 2. The Congruences Let r p =, p throughout this section where p 5 is a prime. Before we prove the congruences a b we will show that p 2p,r pγ 0 for all γ SL 2 Z. For γ = we know by 20 that c d p 2p,r pγ = min gcd2 a + κλc, 2pc + p gcd2 pa + κλc, 2pc λ {0,...,2p } 24 2p 2p 2 = min gcd2 a + κλc, 2pc + p gcd2 a + κλc, 2c λ {0,...,2p } 24 2p 2 = min gcd2 a + κλc, 2p + p gcd2 a + κλc, 2. λ {0,...,2p } 24 2p 2 The last rewriting follows from gcda, c = because ad bc =. Next we will show that p 2p,r p is nonnegative by proving that F a, c, p, λ := gcd2 a + κλc, 2p 2p for all integers a, c, p and λ. We split the proof in four cases: + p gcd2 a + κλc, 2 2 gcda + κλc, 2p = F a, c, p, λ = 2p + p 2 0 gcda + κλc, 2p = 2 F a, c, p, λ = 2 p + 2p 0 gcda + κλc, 2p = p F a, c, p, λ = p 2 + p 2 = 0 gcda + κλc, 2p = 2p F a, c, p, λ = 2p + 2p = 0 Because p 2p,r pγ = min λ {0,...,2p } 24 F a, c, p, λ we know p 2p,r pγ 0. 0

7 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES 7 We are now ready to prove the congruences in Theorem.4. We start with 8: a 5 0n + 2, 6 0 mod 2 We apply Lemma.8. We see that 0, 5, 0, 2, r 5 =, 5. We choose the sequence a δ in Lemma.8 to be the zero sequence this will be so for all the congruences in this paper. Because a δ 0 and because p 0,r 5 0 we see that p 0,r 5γ + p aγ 0 for any γ SL 2 Z. Finally ν = = We choose u = 2 in the lemma and note that c r n = a 5 n for all n 0. Then 8 is true iff a 5 2 a 5 2 a 5 22 a 5 6 a 5 6 a 5 26 mod 2. These values of a 5 are all even as can be seen in the Appendix below, so 8 is proven. A similar approach can be used to prove 9 4. In particular let t p be as in Theorem.7 and r p =, p. Then p 2p, p, p, p, t p, r p. We again set a δ 0 and see as before that for any γ SL 2 Z. We further obtain p 2p,r pγ + p aγ 0 ν = ν p = p 3 2 p 2 2 p + 24 p + 2 p2 48p t p 2p = p 8 p p2 48p t p 2p. Putting these values in a table we obtain p ν p ν p We conclude by Lemma.8 that for all n 0 we have if for 0 n ν p a p 2pn + t 0 mod u, t P 2p,r pt p, a p 2pn + t 0 mod u, t P 2p,r pt p. In particular we choose u = 2 in the case p = 5,, 3, 9 and u = 8 for p = 7, 7, 23. The values of a t n have been calculated in MAPLE for 5 t 23 and are explicitly given in the Appendix below for 5 p 7. The values of a 9 n and a 23 n have been suppressed in the Appendix due to the length of those tables of data. Given that all of these values are congruent to zero modulo 2 or 8, respectively, it is the case that Theorem.4 is proved. 3. Appendix In this section, we demonstrate the exact values of a t n where t is prime, 5 t 7 which are needed to complete the proofs above.

8 8 SILVIU RADU AND JAMES A. SELLERS n a 5n n a 7n n a n n a n n a n

9 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES 9 n a 3n n a 3n

10 0 SILVIU RADU AND JAMES A. SELLERS n a 7n n a 7n

11 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES n a 7n n a 7n n a 7n

12 2 SILVIU RADU AND JAMES A. SELLERS References [] G. E. Andrews and P. Paule. MacMahon s Partition Analysis XI: Broken Diamonds and Modular forms. Acta Arithmetica, 26:28 294, [2] M. Boylan. Congruences for 2 t -core Partition Functions. Journal of Number Theory, 92:3 38, [3] S. H. Chan. Some Congruences for Andrews-Paule s Broken 2-diamond partitions. Discrete Mathematics, 308: , [4] S. Chen. Arithmetical Properties of the Number of t-core Partitions. Ramanujan Journal, 8:03 2, [5] M. D. Hirscchorn and J. A. Sellers. Some Amazing Facts About 4-cores. Journal of Number Theory, 60:5 69, 996. [6] M. D. Hirschhorn and J. A. Sellers. Some Parity Results for 6-cores. Ramanujan Journal. [7] M. D. Hirschhorn and J. A. Sellers. Two Congruences Involving 4-cores. Electronic Journal of Combinatorics, 32:Article R0, 996. [8] M. D. Hirschhorn and J. A. Sellers. On Recent Congruence Results of Andrews and Paule. Bulletin of the Australian Mathematical Society, 75:2 26, [9] L. W. Kolitsch and J. A. Sellers. Elemenatary Proofs of Infinitely Many Congruences for 8-cores. Ramanujan Journal, 32:22 226, 999. [0] S. Radu. An Algorithmic Approach to Ramanujan s Congruences. Ramanujan Journal, 202, [] G. Shimura. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, 97. Research Institute for Symbolic Computation RISC, Johannes Kepler University, A-4040 Linz, Austria, sradu@risc.uni-linz.ac.at Department of Mathematics, Penn State University, University Park, PA 6802, USA, sellersj@math.psu.edu