PARTITIONS AND (2k + 1) CORES. 1. Introduction

Transcription

1 PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES SILVIU RADU AND JAMES A. SELLERS Abstract. In this aer we rove several new arity results for broken k-diamond artitions introduced in 2007 by Andrews and Paule. In the rocess, we also rove numerous congruence roerties for 2k + -core artitions. The roof technique involves a general lemma on congruences which is based on modular forms.. Introduction Broken k-diamond artitions 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 = q 2n q 2k+n q n 3 q 4k+2n k+/2 η2τη2k + τ = q ητ 3 η4k + 2τ, k, where we recall the Dedekind eta function ητ := q 24 q n q = e 2πiτ. In [], Andrews and Paule roved 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 artitions. Since then, a number of authors have rovided roofs of additional congruences satisfied by broken k-diamond artitions. Hirschhorn and Sellers [9] rovided a new roof of the modulo 3 result mentioned above as well as Date: May 26, Mathematics Subject Classification. Primary P83; Secondary 05A7. Key words and hrases. broken k diamonds, congruences, cores, modular forms, artitions. S. Radu was suorted by DK grant W24-DK6 of the Austrian Science Funds FWF.

2 2 S. RADU AND J. A. SELLERS elementary roofs of the following arity 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 aeared in [] as a conjecture while the other three did not. Soon after the ublication of [9], Chan [3] rovided a different roof of the arity results for 2 mentioned above as well as a number of congruences modulo owers of 5. In this aer, we significantly extend the list of known arity results for broken k-diamonds by roving a large number of congruences which are similar to those mentioned above. Indeed, we will do so by roving a similar set of arity results satisfied by certain t-core artitions. A artition is called a t-core if none of its hook lengths is divisible by t. These artitions have been studied extensively by many, esecially thanks to their strong connection to reresentation theory. Numerous congruence roerties are known for t-cores, although few such results are known modulo 2. Such arity results can be found in [7], [6], [0], [8], [2], [4]. In all of these aers, the value of t which was considered was even; in this aer, we rovide a new set of arity results for t-cores wherein t is odd. The generating function for t-core artitions 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.

3 BROKEN k DIAMOND PARTITIONS AND 2k + CORES 3 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+ mod 2 q n 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 we have k 4k+2n+r 0 mod 2 for all n Z a 2k+ 4k+2n+r 0 mod 2 for all n Z. 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 4k+2n+r 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+2n+4k+2v+r n,v Z 0 mod 2. The reverse direction is analogous. bn k mq 4k+2n+m With this motivation, we now state the full list of arity results we will rove in this aer. With the goal of minimizing the notation, we will write ftn + r, r 2,..., r m 0 mod 2

4 4 S. RADU AND J. A. SELLERS to mean that, for each i {, 2,..., m}, ftn + r i 0 mod 2. Theorem.3. For all n 0, n + 2, 6 0 mod 2, 3 4n + 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 roved in [9]. Thanks to Corollary.2, we see that Theorem.3 is roved once we rove the following corresonding 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 rime, 5 23, is reresented in Theorem.4, which hels to exlain why certain families of broken k diamond artitions aear in Theorem.3 and others do not. Our ultimate goal now is to rovide a roof of Theorem.4. We close this section by develoing the machinery necessary to rove this theorem. For M a ositive integer let RM be the set of integer sequences indexed by the ositive divisors δ of M. Let = δ,..., δ k = M be the ositive divisors of M and r RM. Then we will write r = r δ,..., r δk. For s an integer and m a ositive 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.

5 BROKEN k DIAMOND PARTITIONS AND 2k + CORES 5 Definition.5. For m, M N,r δ RM and t {0,..., m } we define the ma : 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 5 be a rime. Let r := r, r =, R. Then.5 P 2,r t = { t 24t 24t = }, t t mod 2, 0 t 2. Proof. First note that 24 δ δr δ = 2 24 Z. Let m = 2. If s s 2 mod m then [s ] 24m t = [s 2 ] 24m t because 2 is 24 an integer. This imlies that.6 P 2,r t = {t t ts + s 2 mod, s S m, 0 t 2 }. 24 We see that.7 P 2,r t mod 2 = {t mod 2}. Next we comute P 2,r t mod. By.6 we know.8 { } P 2,r t mod = t mod t ts + s 2 mod, s S 24 = {t mod 24t s24t mod, s S } { } 24t 24t = t mod =. By.7 and.8 and the Chinese remainder theorem we obtain P 2,r t mod 2 and we obtain the formula.5 by imosing that the elements of P 2,r t lie between 0 and 2. We now use Lemma.6 to comute P 2,r t for = 5, 7,, 3, 7, 9, 23 and t = 2, 7, 2, 2,, 2, below, resectively. 24t = 5, t = 2. We see that = 2 5 =. For t {, 2} we have 24t 5 = and for t {0, 3, 4} we have 24t 5 {0, }. This imlies that P 0,r 52 {, 2} mod 5. Since t 0 mod 2 we have that

6 6 S. RADU AND J. A. SELLERS P 0,r 52 0 mod 2. Hence by Lemma.6 we have P 0,r 52 = {2, + 5} = {2, 6}. 24t = 7, t = 7. We see that = 7 =. We see that for t {0, 2, 6} we have 24t 7 = and this is all t with this roerty so P 4,r 7 {0, 2, 6} mod 7. Because t mod 2 we obtain by Lemma.6 P 4,r 77 = {0 + 7, 2 + 7, 6 + 7} = {7, 9, 3}. =, t = 2. Here 24t = 5 2 =. We see that for t {, 2, 3, 5, 8} we have 24t = so P 22,r 2 = { +, 2, 3 +, 5 +, 8} = {2, 8, 2, 4, 6}. Similarly we get by Lemma.6 and P 26,r 32 = {2, 0, 6, 8, 20, 22}, P 34,r 7 = {, 5, 7, 9, 25, 27, 29, 33}, P 38,r 92 = {2, 8, 0, 20, 24, 28, 30, 32, 34} P 46,r 23 = {, 5, 2, 23, 29, 3, 35, 39, 4, 43, 45}. We see immediately from the above that Theorem.4 is equivalent to the following theorem. Theorem.7. Let t : {5, 7,, 3, 7, 9, 23} {2, 7, } with t 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, rime with 5 23, and t P 2,r t, we have.9 a 2n + t 0 mod 2 i, where i = { if = 5,, 3, 9, 3 if = 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 a rime, m N and t {0,..., m } we are concerned with roving congruences of the tye c r mn + t 0 mod, n N. The congruences we are concerned with here have some additional structure; namely a r mn + t 0 mod, n 0, t P m,r t. In other words a congruence

7 BROKEN k DIAMOND PARTITIONS AND 2k + CORES 7 is a tule r, M, m, t, with r RM, m, t {0,..., m } and a rime such that a r mn + t 0 mod, n 0, t P t. Throughout when we say that a r mn + t 0 mod we mean that a r mn + t 0 mod for all n 0 and all t P t. The urose of this aer is show the congruences a 2n + t 0 mod 2 when = 5, 7,, 3, 7, 9, 23 and t = 2, 7, 2, 2,, 2,. In order to accomlish our goal we need a lemma [, Lemma 4.5]. We first state it and then exlain the terminology. Lemma.8. Let u be a ositive 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 comlete set of reresentatives of the double coset Γ 0 N\Γ/Γ. Assume that m,r γ i + aγ i 0, i {,..., n}. Let t min := min t P m,rt t and ν := a δ + 24 δ N δ M Then if ν r δ [Γ : Γ 0 N] δ N δa δ c r mn + t q n 0 mod u for all t P m,r t then c r mn + t q n 0 mod u for all t P m,r t. 24m δ M δr δ t min m. The lemma reduces the roof 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 tule m, M, N, r δ, t belongs to iff m, M, N, r δ RM, t {0,..., m }; m imlies N for every rime ; δ M imlies δ mn for every δ such that r δ 0; κn δ M r δ mn δ 0 mod 24; κn δ M r δ 0 mod 8;

8 8 S. RADU AND J. A. SELLERS 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 grous Γ, Γ 0 N and Γ : { } a b Γ := a, b, c, d Z, ad bc =, c d Γ 0 N := for N a ositive integer, and Γ := { a b c d { h 0 } Γ N c } h Z. For the index we have [Γ : Γ 0 N] := N N + see, for examle, [2]. a b Finally for m, M, and r RM and γ = we define c d.20 m,r γ := min λ {0,...,m } 24 and 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 =, throughout this section where 5 is a rime. Before we rove the congruences we will show that 2,r γ 0 for all γ SL 2 Z. a b For γ = we know by.20 that c d 2,r γ = min gcd2 a + κλc, 2c + gcd2 a + κλc, 2c λ {0,...,2 } = min gcd2 a + κλc, 2c + gcd2 a + κλc, 2c λ {0,...,2 } = min gcd2 a + κλc, 2 + gcd2 a + κλc, 2. λ {0,...,2 } The last rewriting follows from gcda, c = because ad bc =. Next we will show that 2,r is nonnegative by roving that F a, c,, λ := gcd2 a + κλc, gcd2 a + κλc, 2 2 0

9 BROKEN k DIAMOND PARTITIONS AND 2k + CORES 9 for all integers a, c, and λ. We slit the roof in four cases: gcda + κλc, 2 = F a, c,, λ = gcda + κλc, 2 = 2 F a, c,, λ = gcda + κλc, 2 = F a, c,, λ = = 0 gcda + κλc, 2 = 2 F a, c,, λ = = 0 Because 2,r γ = min λ {0,...,2 } F a, c,, λ we know 24 2,r γ 0. We are now ready to rove the congruences in Theorem.4. We start with.8: a 5 0n + 2, 6 0 mod 2 We aly 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 aer. Because a δ 0 and because 0,r 5 0 we see that 0,r 5γ + 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 Aendix below, so.8 is roven. A similar aroach can be used to rove.9.4. In articular let t be as in Theorem.7 and r =,. Then 2,, , t, r. We again set a δ 0 and see as before that 2,r γ + aγ 0 for any γ SL 2 Z. We further obtain ν = ν = t 2 = t 2. Putting these values in a table we obtain

10 0 S. RADU AND J. A. SELLERS ν ν We conclude by Lemma.8 that for all n 0 we have if for 0 n ν a 2n + t 0 mod u, t P 2,r t, a 2n + t 0 mod u, t P 2,r t. In articular we choose u = 2 in the case = 5,, 3, 9 and u = 8 for = 7, 7, 23. The values of a t n have been calculated in MAPLE for 5 t 23 and we confirm that they satisfy the desired congruences. The authors would be hay to suly this data to anyone interested. Given that all of these values are congruent to zero modulo 2 or 8, resectively, it is the case that Theorem.4 is roved. 3. Acknowledgements The authors thank the referee of this aer for carefully reading the original manuscrit and suggesting valuable changes to the aer. Since the submission of this article, the authors have determined that.8,., and.2 of Theorem.4 aear in an alternate form in Garvan s work [5]. However, it should be noted that the roof technique of Garvan is different from the technique in this aer, although both rely significantly on modular forms. Our belief is that our results rovide a unified treatment of arity results for t-cores where t is a rime, 5 t 23. 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 artitions. Discrete Mathematics, 308: , [4] S. Chen. Arithmetical Proerties of the Number of t-core Partitions. Ramanujan Journal, 8:03 2, [5] F. Garvan. Some congruences for artitions that are -cores. Proceedings of the London Mathematical Society, 663: , 993.

11 BROKEN k DIAMOND PARTITIONS AND 2k + CORES [6] M. D. Hirschhorn and J. A. Sellers. Some Amazing Facts About 4-cores. Journal of Number Theory, 60:5 69, 996. [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. Some Parity Results for 6-cores. Ramanujan Journal, 3:28 296, 999. [9] M. D. Hirschhorn and J. A. Sellers. On Recent Congruence Results of Andrews and Paule. Bulletin of the Australian Mathematical Society, 75:2 26, [0] L. W. Kolitsch and J. A. Sellers. Elemenatary Proofs of Infinitely Many Congruences for 8-cores. Ramanujan Journal, 32:22 226, 999. [] S. Radu. An Algorithmic Aroach to Ramanujan s Congruences. Ramanujan Journal, 202:25 25, [2] G. Shimura. Introduction to the Arithmetic Theory of Automorhic Functions. Princeton University Press, 97. Research Institute for Symbolic Comutation RISC, Johannes Keler University, A-4040 Linz, Austria, sradu@risc.uni-linz.ac.at Deartment of Mathematics, Penn State University, University Park, PA 6802, USA, sellersj@math.su.edu