ON THE BROKEN 1-DIAMOND PARTITION

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1 ON THE BROKEN -DIAMOND PARTITION ERIC MORTENSON Abstract. We introduce a crank-ike statistic for a different cass of partitions. In [AP], Andrews and Paue initiated the study of broken k-diamond partitions. Their study of the respective generating functions ed to an infinite famiy of moduar forms, about which they were abe to produce interesting arithmetic theorems and conjectures for the reated partition functions. Here we estabish a crank-ike statistic for the broken -diamond partition and discuss its roe in congruence properties.. Introduction and Statement of Resuts Among the most ceebrated resuts in the theory of partitions are Ramanujan s congruences for the partition function: p(5n (mod 5, p(7n (mod 7, p(n (mod. We note that there are many different proofs and generaizations of these formuas. Dyson [D] provided combinatoria insight into these congruences with a simpe statistic caed the rank. Athough this expains the first two congruences (see Atkin and Swinnerton-Dyer [AS], it does not expain the third. For the third, Dyson conjectured the existence of an additiona statistic, which he caed the crank. Forty years after his conjecture was made, Andrews and Garvan ([G], [AG] defined a function and showed that it does indeed dissect the Ramanujan congruences moduo ; it aso expains the moduo 5 and 7 congruences. As this paper shows, crank-ike statistics exist in other partition settings as we. In [AP] Andrews and Paue, using MacMahon s Partition Anaysis, initiated the study of broken k-diamond partitions. Their study of the respective generating functions ed to an infinite famiy of moduar forms, about which they produced interesting arithmetic theorems and conjectures for the reated partition functions. In this paper, we introduce a new crankike statistic for this cass of partitions. More specificay we consider the broken -diamond partition, and motivated by works of Ahgren and Ono ([O], [AO] and Mahburg [M], we discuss partition congruences associated to this statistic. To introduce broken -diamond partitions, we foow the exposition in [AP], and begin with asic exampe of cassica pane partitions, treated by MacMahon in [Ma]. Here, the nonnegative integer parts a i of the partitions are paced at the corners of a square such that the 2000 Mathematics Subject Cassification. P8, 05A7, F. The author thanks the Nationa Science Foundation for their generous support.

2 2 ERIC MORTENSON foowing order reations are satisfied: (. a a 2, a a, a 2 a 4, and a a 4. We use arrows as an aternative description for reations; for instance, Fig. represents the reations (.. We interpret an arrow pointing from a i to a j as a i a j. a a a 4 a 2 Figure. The inequaities (. In [Ma], MacMahon derives the genera generating function using partition anaysis (.2 ϕ : x a xa 2 2 xa xa 4 4 x 2 x 2x ( x ( x x 2 ( x x ( x x 2 x ( x x 2 x x 4, the sum being over a non-negative integers a i satisfying (.. He further observes that by setting x x 2 x x 4 q, the resuting generating function is ( q( q 2 2 ( q. As in [AP], we now consider the pane partition diamond of ength n. a a 6 a 9 a n a... a n+ a 4 a 7 a 0 a n 2 a 2 a 5 a 8 a n Figure 2. A pane partition diamond of ength n Definition.. For n define H n : {(a,...,a n+ N n+ : the a i satisfy the reations in Fig. 2}, h n : h n (x,...,x n+ : x a xa 2 2 xa n+ (a,...,a n+ H n and h n (q : h n (q,...,q. n+, In [AP], they prove a generating function in cosed form which speciaizes to

3 Theorem.2. For n, ON THE BROKEN -DIAMOND PARTITION h n (q n j0 ( + qj+2 n+ j ( qj. Pane diamond partitions with a source deeted are then considered. Definition.. For n define H n : {(a 2,...,a n+ N n : the a i satisfy the reations in Fig. 2 where the vertex abeed a has been deeted}, and h n(q : h n(q,q,...,q. h n : h n(y 2,...,y n+ : (a 2,...,a n+ H n Again, they prove a generating function which speciaizes to Theorem.4. For n, h n(q n j0 ( + qj+ n j (. qj y a 2 2 ya ya n+ n+, Combining these two notions, roken -diamond partition consists of two separate pane partition diamonds of ength n, where in one of them the source is deeted. b n b 6 b a a n b n a n+ b 7 b 4 a a 4 b n b 5 b 2 a 2 a n Figure. A broken -diamond of ength 2n Definition.5. For n define H n : {(b 2,...,b n+,a,a 2,...,a n+ N 6n+ : the a i and b i satisfy a the reations in Fig. }, h n : h n(y 2,...,y n+ ;x,x 2,...,x n+ : y b 2 2 yb n+ n+ xa xa 2 2 xa n+ n+, and h n(q : h n(q,q,...,q. (b 2,...,b n+,a,a 2,...,a n+ H n One sees that h n h nh n, and given Theorems.2 and.4, with n, we have

4 4 ERIC MORTENSON Theorem.6. h j ( + q j ( q j 2 ( + q j q 6 η(2τ η ( τ η(τ η ( 6τ, where q e 2πiτ, and η(τ : q 24 n ( qn is Dedekind s η-function. Definition.7. For n 0, et (n denote the tota number of broken -diamond partitions, (nq n : h. n0 To state our resuts, more definitions are required. We define the foowing: (. D : {p prime : p,25,7,47,59, or 8 (mod 84}, (.4 δ : 2 6, ε : ( 6, (.5 S : {β {0,,..., } : ( β+δ 0 or ε }. Motivated by Ahgren and Ono [AO], we show that for each prime D there is a Ramanujantype congruence. Theorem.8. Suppose that D, k a positive integer, and β S. Then a positive proportion of primes Q (mod 6 have the property that, Qn + 0 (mod k 6 for a n 6β (mod 6 that are not divisibe by Q. Coroary.9. Suppose that D, k a positive integer, and β S. Then there are infinitey many non-nested arithmetic progressions {An + B} {n + β} such that for every integer n we have (An + B 0 (mod k. We discuss a statistic and its roe in the congruence properties of (n. In [AP], a straightforward proof of the foowing theorem is given. Theorem.0. If n 0, then (2n + 0 (mod. Given λ (b 2,b,b 4,... a,a 2,a,... H, we now define a new statistic R to expain this congruence. Definition.. (.6 R(λ : [ a i0 ] [ max(a i+ a i+2,0 b 2 j ] max(b j+ b j+2,0. The number of partitions of n with statistic m is denoted by R(m,n, and the number of partitions of n with statistic congruent to m moduo N by R(m,N,n. This statistic then provides a combinatoria proof of Theorem.0. Theorem.2. For n 0, R(0,,2n + R(,,2n + R(2,,2n +.

5 ON THE BROKEN -DIAMOND PARTITION 5 Motivated by Mahburg [M], we can show that for each prime D, there is Ramanujan-type congruence expained by this statistic. Theorem.. Suppose that D, k and j positive integers, and β S. Then a positive proportion of primes Q (mod 6 have the property that for every 0 m j, ( R m, j, Qn + 0 (mod k 6 for a n 6β (mod 6 that are not divisibe by Q. The foowing two coroaries are immediate. Coroary.4. Suppose that D, k and j positive integers. Then there are infinitey many non-nested arithmetic progressions An + B such that for every 0 m j, R(m, j,an + B 0 (mod k. Coroary.5. Suppose that D, and k a positive integer. Then there are infinitey many non-nested arithmetic progressions An + B such that the statistic provides a proof of the congruence (An + B 0 (mod k. In Section 2, in the spirit of the dissections for the generating functions of the crank and rank found in Ramanujan s Lost Notebook, as shown in [G], we prove Theorem.2. In Section, we provide preiminaries on moduar forms and Kein forms needed in the proofs of Theorems.8 and.. These proofs are found in Sections 4 and 5, respectivey. The proofs of two technica propositions are found in Section 6. Acknowedgements The author woud ike to thank George Andrews for his suggestion to ook for a statistic to expain Theorem.0, Kar Mahburg and Scott Ahgren for their hepfu comments in preparing this manuscript, and the referee for his thorough and timey reading of this paper. Part of this research was carried out during a stay at the Max-Panck-Institut für Mathematik in Bonn, Germany, the author thanks this institution for its support. 2. The Statistic and its Generating Function We begin by stating the cosed forms of the generating functions that speciaize to Theorems.2 and.4, respectivey. Theorem 2.. Let X 0 :, and X m : x x 2 x m, (m. For n, h n (x,x 2,...,x n+ n+ j Theorem 2.2. Let Y : and Y n : y 2 y...y n. For n 2, h n(y 2,y,...,y n+ n+ j2 n X i+ X i+. X j X i0 i+ x i+ n Y j i0 Y i+ Y i+ Y i+ y i+.

6 6 ERIC MORTENSON These generating functions are now used to estabish a generating function for the statistic: Theorem 2.. We have the foowing generating function for the statistic R(m,n: n0 m R(m,nz m q n n 0 ( zq 6n+4 ( z q 6n+2 ( zq n+ ( q n+2 ( z q n+ ( q n+. Proof of Theorem 2.. To expain the first set of brackets in Definition., it suffices to examine the foowing factor from Theorem 2.: zx i+ X i+ ( zx i+2 ( X i+ x i+ (zx i+ x i+2 k zx2 i+ x i+2x i+ X i+ x i+ k0 { } z k X k X i+ x i+x k i+2 z k+ Xi+ k+2 xk+ i+2 x i+ i+ k0 { + X i+ x i+ k0 k0 } z k Xi+x k k i+2( X i+ x i+ k Xi+x k k i+ + z k Xi+2. k k In other words, the z is effectivey counting the contribution to the argest part in terms of /( zx k, whie ignoring the contribution from /( X i+ x i+. In a simiar fashion, the anaogous factor from Theorem 2.2 expains the second set of brackets in Definition.. The generating function for the statistic foows. We define (2. F(z,q : (zq 4 ;q 6 (z q 2 ;q 6 (zq;q (z q;q (q 2 ;q (q;q, where this is the right hand side of the equation in Theorem 2., and we use Theorem 2. to obtain the foowing dissection. Theorem 2.4. Let ρ e 2πi/, then F(ρ,q A 4,4 (q 6 ρq 4 A 4, (q 6 ρ 2 q 2 A 4,2 (q 6, where A k,i (q : n 0,±i (mod 2k+ q n. We reca the notation for Ramanujan s Theta function (p., [AB]: (2.2 f(a,b : n Za n(n+/2 b n(n /2 ( a;ab ( b;ab (ab;ab.

7 ON THE BROKEN -DIAMOND PARTITION 7 A simpe speciaization gives, (2. ( n q an2 +bn (q 2a ;q 2a (q b+a ;q 2a (q b+a ;q 2a. n The foowing two emmas prove the above theorem. Lemma 2.5. Let ρ e 2πi/. Then F(ρ,q (ρq 4 ;q 6 (ρ q 2 ;q 6. Proof. This foows from a series of basic q-series manipuations. F(ρ,q (ρq 4 ;q 6 (ρ q 2 ;q 6 (ρq;q (ρ q;q (q 2 ;q (q;q (ρq 4 ;q 6 (ρ q 2 ;q 6 (ρq;q (ρ q;q (q 2 ;q (q;q (q;q (q;q Lemma 2.6. Let ρ e 2πi/. Then (ρq4 ;q 6 (ρ q 2 ;q 6 (q;q (q ;q (q 2 ;q (q;q (ρq 4 ;q 6 (ρ q 2 ;q 6. (ρq 4 ;q 6 (ρ q 2 ;q 6 A 4,4 (q 6 + ρq 4 A 4, (q 6 + ρ 2 q 2 A 4,2 (q 6. Proof. With Ramanujan s Theta function notation, we have (ρq 4 ;q 6 (ρ q 2 ;q 6 (ρq 4 ;q 6 (ρ q 2 ;q 6 (q 6 ;q 6 (q 6 ;q 6 (q 6 ;q 6 ( ρq 4 n(n+/2 ( ρ q 2 n(n /2 (q 6 ;q 6 n n ( n ρ n q n2 +n (q 6 ;q 6 n 2 k0 Substituting in the vaues of k, we obtain (ρq 4 ;q 6 (ρ q 2 ;q 6 { (q 6 ;q 6 ( n q n(9n+ ρ n ρ k ( n q (n+(9n+4 + ρ 2 With (2., and more cacuation, we obtain the desired resut. n ( n+k q (n+k(9n+k+. n Proof of Theorem.2. From Theorem 2.4 it is immediate that 2 R(m,2n + ρ m R(k,,2n + ρ k 0, m k0 ( n q (n+2(9n+7}. since F(z,q is supported ony on even powers of q. The eft-hand side is poynomia in ρ e 2πi/ over Z. Because the minima poynomia for ρ over Q is p(x + x + x 2, it foows that R(0,,2n + R(,,2n + R(2,,2n +.

8 8 ERIC MORTENSON. Preiminaries for the Proofs of Theorems.8 and... Moduar Forms Background. The reader can refer to [K] and [O] for basic facts about moduar forms. We note that there are severa ways to produce new moduar forms from od ones. Among them are taking a twist, appying the Hecke Operator, and restricting the q-expansion to terms that ie in certain arithmetic progressions. Definition.. Suppose that f(τ n0 a(nqn M k (Γ 0 (N,χ, and that ψ is a Dirichet character. Then the twist of f by ψ is (. (f ψ(τ : n 0 ψ(na(nq n. Definition.2. Suppose that f(τ n0 a(nqn M k (Γ 0 (N,χ, and p N prime. Then the action of the p-th Hecke Operator Tp k,χ : T(p is given by (.2 f(τ T(p : n 0(a(pn + χ(pp k a(n/pq n. We aso have the foowing basic properties: Proposition.. Suppose that f(τ M k (Γ 0 (N,χ, and p N prime. Then the action of T(p is space-preserving, i.e., If ψ is a character with moduus M, then f(τ T(p M k (Γ 0 (N,χ. (f ψ(τ M k (Γ 0 (NM 2,χψ 2. We note that the twist of a moduar form by a quadratic character can be written in terms of the sash operator. (We surpress the weight term in the sash operator as the expression is independent of it. If p is a prime, where the Gauss sum g : g p p v (v p e2πiv/p, we can write (see p. 28 [K] (. f(τ ( g p p p v ( v p f(τ v/p. 0 Discarding coefficients that ie in certain arithmetic progressions aso produces moduar forms. Proposition.4. Suppose that f(τ n a(nqn S k (Γ (N, where k is an integer. If t and 0 r t, then a(nq n S k (Γ (Nt 2. n r (mod t In Section 5 we wi need to simutaneousy find congruences for moduar forms of differing weights and eves. The foowing theorem is based on modifictions by Ahgren and Ono [AO], [O2], and Mahburg [M] to a cassica resut due to Deigne and Serre on Gaois representations associated to moduar forms.

9 ON THE BROKEN -DIAMOND PARTITION 9 Theorem.5. Suppose that k i and N i are integers, and that χ i is a Dirichet character for i r. Let g (τ,...,g r (τ be integer weight moduar forms with agebraic coefficients such that g i (τ M ki (Γ 0 (N i,χ i. If M, then a positive proportion of primes p (mod N N r M have the property that for every i, g i (τ T(p 0 (mod M..2. Siege and Kein Forms. Kubert and Lang [KL] studied transformation properties for certain moduar functions which generaize η 2 (τ. These generaizations, known as Kein and Siege forms, are vita in studying the generating function of our statistic. Here we write (.4 q q τ e 2πiτ, q z e 2πiz, and z a τ + a 2. Definition.6. Let (a,a 2 R 2, ( The (a,a 2 -Siege function has the q-expansion g (a,a 2 (τ : q (/2B 2(a τ e 2πia 2(a /2 ( q z n ( q z qτ n ( q where B 2 (X X 2 X + 6 is the second Bernoui poynomia. (2 The (a,a 2 -Kein form is given by t (a,a 2 (τ : i 2π g(a,a 2 (τ η 2. (τ We have the foowing basic transformation properties (pp [KL]. Proposition.7. If α SL 2 (Z, then t (a,a 2 (ατ t (a,a 2 α(τ. Proposition.8. Let (a,a 2 R 2 and (b,b 2 Z 2. Writing (a + b,a 2 + b 2 (a,a + (b,b 2, we have t (a +b,a 2 +b 2 (τ t (a,a 2 +(b,b 2 (τ ε((a,a 2,(b,b 2 t (a,a 2 (τ, where ε((a,a 2,(b,b 2 has absoute vaue, and is given expicity by ε((a,a 2,(b,b 2 ( b b 2 +b +b 2 e 2πi(b 2a b a 2. z qn τ, Letting the overbar denote reduction (mod N, the above two propositions show, Coroary.9. If Γ 0 (N and 0 s N, then t (0,s/N (τ β t (0,ds/N (τ, where β : e ( cs+(ds ds 2N + cs(ds ds cs ds 2N. 2 The foowing coroary describes a genera case in which the mutipier β is trivia. Here M! denotes the space of weaky hoomorphic forms. Coroary.0. Let 0 r,s N, then t (r/n,s/n (τ M! (Γ (2N 2.

10 0 ERIC MORTENSON 4. Proof of Theorem.8 The proof borrows techniques from [AO]. We first estabish a key theorem, in that we construct cusp forms whose coefficients capture the reevant vaues of (n. We reca the defintions of D, ǫ, δ and S in (., (.4, and (.5. Theorem 4.. Suppose D and that m is a positive integer. If β S, then there is an integer λ,m and a moduar form F,m,β (τ S λ,m (Γ (08 5 Z[[q]] such that F,m,β (τ (n + βq 8n+8β (mod m. n0 The cusp form wi be the product of two moduar forms, one vanishing at a cusps a/c where c, the other vanishing at a a/c where c. The atter is where we need D. The foowing usefu proposition can be shown viasic facts about moduar forms. Proposition 4.2. Let 5 be a prime. We define E,t (τ : ηt (τ η( t τ. (4. ( E,t (τ M t 2 (Γ 0 ( t,χ,t, where χ,t ( : ( ( t 2 t. (2 If a, 0 b < t, then ord a/ b(e,t > 0, i.e. E,t (τ vanishes at those cusps of Γ 0 ( t not equivaent to. ( E,t (τ m (mod m. Define η(2τη(τ : g(τ η(τ η(6τ (nq n 6, If 5 is prime, one can show (4.2 f (τ a (nq n : g (τ g(τ This impies (4. n ( a (nq n n Further, we can show n0 (nq n+δ n0 M (Γ 0 (6,χ triv. n ( q n ( q 6n ( q 2n ( q n. (4.4 f (τ : f (τ ε f ( (τ ( ε ( n a (nq n M (Γ 0 (6,χ triv. n We want to show that the quotient f (τ/g (τ vanishes at a cusps a/c where c, D. We subdivide those cusps into four groups and define v i to be the order of vanishing of f (τ at any cusp of a/c C i, and v i that of g (τ. ( C : { a c : 6 c}, v ( , v ( (2 C 2 : { a c : c, 2 c}, v 2 ( 2 24, v 2 ( 2 24, ( C : { a c : 2 c, c}, v ( , v ( ,

11 ON THE BROKEN -DIAMOND PARTITION (4 C 4 : { a c : c; 2, c}, v 4 ( , v 4 ( Next we have the foowing proposition whose proof we defer to Section 6. Proposition 4.. Given a c C i, i 4, choose b,d such that SL 2 (Z, and define α by f α q v i +..., then f ( ( vi α q v i For D, quadratic reciprocity gives us, ( v ( v2 ( v ( v4 (4.5 ε. We note that for a/c C i, and b,d chosen such that SL 2 (Z, (4.6 g q (τ v i Equation (4.4 and Proposition 4. yied, Proposition 4.4. Given D and a cusp a/c such that c, ( f (τ (4.7 ord a/c g v i + v i (τ > 0. f Proof of Theorem 4.. We consider (τ g (τ E,(τ m. By Propositions 4.2(2 and 4.4, with m sufficienty arge, this vanishes at a the cusps. By (4., (4.4 and Propositions 4.2(, 4.4, we have f (τ g (τ E,(τ m (4.8 (n δ q n 2 6 n 0 (mod + 2 ( n ε (n δ q n 2 6 (mod m. It foows that f (8τ g (8τ E,(8τ m is a cusp form on Γ 0 (8 6, and we then appy Proposition.4. Proof of Theorem.8. Fix D, an integer β S, and write (4.9 F,m,β (τ a,m,β (nq n n n 8β (mod 8 n + q n (mod m. 8 By Theorem.5, for a positive proportion of primes Q (mod 8, (4.0 F,m,β (τ T(Q 0 (mod m.

12 2 ERIC MORTENSON Theorem.8 then foows from the foowing. We see that if n 8β (mod 8, that Qn 8β (mod 8; and if gcd(q,n, we can use (.2 to obtain Qn + (4. 0 a,m,β (Qn (mod m Proof of Theorem. We borrow techniques from [AO] and [M], and begin by producing a generating function for R(m,N,n in terms of Kein forms. We again remind ourseves of the defintions of D, ǫ, δ and S in (., (.4, and (.5. Recaing (2., setting N : j, ζ : e 2πi/N, λ a partition, and etting 0 s N, we examine the sum (5. which eads to (5.2 N F(ζ s,zζ ms N N s0 λ R(m,N,nq n N n 0 N s0 ζ R(λ s ms q λ λ q λ N ζs(r(λ m R(m,N,nq n, n 0 N s (ζ s q 4 ;q 6 (ζ s q 2 ;q 6 ζ ms (ζ s q;q (ζ s q;q (q 2 ;q (q;q + (nq n. N We reca (4., Definition.6, and make the substitution ζ ζ to obtain (5. R(m,N,nq n n 0 N s n 0 h s q 6 g(τ t(/,s/n(6τ η (6τ t (0,s/N (τ η(2τ ζms N + N where h s : ζ s/2 ( ζ s /ζ s/6. Now we define a function more amenabe to being anayzed by moduar forms: ( g m (τ : N R(m,N,nq n+δ ( q n ( q 6n (5.4 ( q 2n ( q n n 0 N s n (nq n, n 0 g (τ g(τ hst (/,s/n (6τ η (6τ t (0,s/N (τ η(2τ + g (τ g(τ, with P m (τ and P(τ denoting the two summands in the fina expression. As in the previous section, we proceed to reate g m (τ to cusp forms by first subdividing the cusps a/c, N c into four groups. ( C : {a c : 6N c}, (2 C 2 : {a c : N c, 2 c}, ( C : {a c : 2N c, c}, (4 C 4 : {a c : N c; 2, c}.

13 ON THE BROKEN -DIAMOND PARTITION The foowing proposition wi be proved in Section 6: Proposition 5.. Given s N, a c C i, pick b,d with SL 2 (Z. Define α s by g (τ t (/,s/n (6τ η (6τ α g(τ t (0,s/N (τ η(2τ s q v i Then g (τ t (/,s/n (6z η (6τ ( g(τ t (0,s/N (τ η(2τ where the v i s are as in the previous section. ( vi α s q v i +..., With this and previous arguments, we prove the foowing proposition, which provides us with a theorem crucia to the proof of Theorem.. We reca the tide notation from (4.4 for the foowing theorem and proposition. Proposition 5.2. Let D. If k is sufficienty arge, there exist integers λ, λ and some Dirichet character χ such that P(8τ ( g (8τ E j+(8τ k S λ (Γ 0 (08 max{,j+},χ (2 P m(8τ g (8τ E j+(8τ k S λ (Γ (944 2 N 2. Proof of Proposition 5.2. For (, reca that P(τ g (τ g(τ M (Γ 0 (6,χ triv and then modify the arguments of the previous section. For the (2, we know that E j+ (τ vanishes at each cusp a/c, N c. Once k is taken to be sufficienty arge, it ony remains to show that P m (τ/g (τ vanishes at each cusp a/c with N c, but this foows from Proposition 5., equation (4.5, and quadratic reciprocity. To determine the appropriate eve of the congruence subgroup, basic facts about moduar forms and Coroary.0 yied P m (τ M!(Γ (2 2 4 N 2, and the rest is straightforward. Combining the above proposition with the definition of g m (τ in (5.4 and the congruence properties from Proposition 4.2, we obtain the foowing theorem. This is essentiay heading in the same direction as that of Theorem 4.; here, however, we work with two moduar forms. Theorem 5.. For k 0 and 0 m N there is a character χ, positive integers λ and λ, and moduar forms ( F(τ S λ (Γ 0 (08 max{,j+},χ, and (2 F m (τ S λ (Γ (944 2 N 2, such that gm(8τ g (8τ F m(τ + F(τ (mod k. Proof of Theorem.. By (4.4 and (5.4, we have g m (8τ g (8τ (5.5 N R(m,N,n δ q 8n 2 n 0 (mod + 2 ( n ε N R(m,N,n δ q 8n 2

14 4 ERIC MORTENSON Restricting the above sum to those indices n β + δ (mod, β S, yieds a new series where λ β equas if β δ and equas 2 otherwise. g m,β (τ : λ β N R(m,N,n δ q 8n (5.6 2 n β+δ +n λ β N R(m,N,n + βq 8β+8δ +8n 2 λ β n 0 n 8β (mod 8 ( N R m,n, n + q n 8 However, upon examining Theorem 5., we can restrict the q-expansions of F m (τ and F(τ to those indices with n β + δ (mod to obtain (5.7 g m,β (τ F m,β (τ + F β (τ (mod k, and the foowing from Proposition.4: (5.8 (5.9 F m,β (τ S λ (Γ (944 4 N 2, F β (τ S λ (Γ 0 (08 2+max{,j+},χ. Theorem.5 provides a positive proportion of primes Q (mod 8 such that (5.0 F m,β (τ T(Q F β (z T(Q 0 (mod k for a m. This in turn gives that g m,β (τ T(Q 0 (mod k for a m. Arguing as in the proof of Theorem.8 we obtain for a n 8β (mod 8, gcd(n, Q that ( λ β N R m,n, Qn + 0 (mod k. 8 Theorem. foows by noting that because N is fixed and k is arbitrary, dividing by N proves the congruences. 6. Proofs of Propositions 4.4 and Proof of Proposition 4.4. We( restrict ourseves to the case 2N c, c, the other three cases being simiar. Let δ Z +, SL 2 (Z, and (, denote the greatest common divisor, then we can write δa δb δa/(c,δ γδ,b (c,δ Bδ (6., c/(c,δ γ δ,d 0 δ/(c,δ ( δa δb δa/(c,δ γ δ,b (c,δ Bδ (6.2 c/(c,δ γ δ,d, 0 δ/(c,δ where the eft matrix in each product is in SL 2 (Z, (6. γ δ,d : γ δ,d, γ δ,b : γ δ,b, and B δ γ δ,d δ b γ δ,b d. The transformation formua for Dedekind s eta function (see p. 6 [R] yieds, η (δτ ( δa/(c,δ (6.4 e 2πi 24 (2 (c,δ δ B δ q (c,δ 2 24 δ ( η(δτ

15 ON THE BROKEN -DIAMOND PARTITION 5 It then foows (6.5 f (τ α q ov +... where (6.6 α : e 2πi { (c, 24 (2 B (c,2 2 B 2 (c, B + (c,6 } 6 B 6, ov : 24 (2 { (c,2 (c,22 2 (c,2 + (c,62 6 }. We note ov v. By recaing how γ δ,b, γ δ,d, and B δ were defined, (6.7 ( a γ,b c γ,d ( B 0, 6a 6b and by setting γ 6,d : γ,d, and γ 6,b : 2γ,b, we can reate B and B 6 : a γ6,b 2 B6, c/2 γ 6,d 0 (6.8 B 6 γ 6,d 6 b γ 6,b d 2 B. With this and the fact that 24 2, the expression for α simpifies to (6.9 α e 2πi 24 { } (2 B e 2πi 24 (2 We reca expression (., define γ ν : { γ,b } d. ν/, and choose ν 0 d 2 ν (mod : (6.0 f ( (τ g g g ν0 ν0 ( ν [ ] [ f ] γ ν ( ν [ ] f γ ν γ ν γ ν ( ν [ f γ ν γ ] [ ] ν γ ν. ν0 By choice of ν, ( (6. γ ν γ a cν/ b + (ν ν a νd/ cνν / 2 ( a b + cν : / c d, and thus define a, b, c, d. Using (6.5 and (6.9, (6.2 f ( (τ g ν0 ( ν e 2πi 24 (2 { γ,b d 4 (ν } q v +....

16 6 ERIC MORTENSON Considering the anaog of (6. for (6., one sees that γ,b γ,b (mod. Couped with the definition d : d + cν / we have (6. f ( (τ g ( ν e 2πi 24 (2 g ν0 ν0 { γ,b d ( ν α e 2πi 24 (2 } e 2πi 24 (2 { γ,b c 4 { γ,b c 4 } ν q v } ν q v +... Recaing (6., we note a γ,d γ,b c, and therefore γ,b c (mod. From this it is cear that γ,b c 4 Z and that f ( (τ g ( 2πi g 24 (2 { γ,b c 4 } (6.4 αq v +... g2 ( ( v ( αq v v + αq v Proof of Proposition 5.. We examine the case 2N c, c. For the sake of exposition we further refine it to the subcase 4 c, with the other cases being simiar. We compute the ead term for each of P m,s (τ and P m,s ( (τ, where from Section 5, (6.5 P m,s (τ : g (τ g(τ h s t (/,s/n (6τ t (0,s/N (τ η (6τ η(2τ. We reduce the probem to finding the ead terms for each of the expansions: g (τ,t g(τ (/,s/n (6τ,, and η (6τ. t (0,s/N (τ η(2τ From the previous subsection: g (τ (6.6 g(τ By Coroary.9, (6.7 t (0,s/N (τ e 2πi 24 (2 ( γ,b d q 24 ( β s t (0,ds/N (τ β s ( i 2π ω q ds Using the transformation aw for Dedekind s eta function, η (6τ ( c/2 (6.8 e πi 2 {4ac} q η(2τ For the Kein form, we require more work. Note, 6a 6b a γ6,b 2 B6 (6.9, c/2 γ 6,d 0

17 ON THE BROKEN -DIAMOND PARTITION 7 where the eft matrix in the product is in SL 2 (Z. By Propositions.7 and.8, ( a γ6,b (6.20 t (/,s/n (6τ t (/,s/n ( c/2 γ 2 τ + B 6 6,d t a+ cs 2N,γ 6,b +γ 6,d s ( 2 τ + B 6 t (a,a 2 +(b,b 2 ( 2 τ + B 6 N ε((a,a 2,(b,b 2 t (a,a 2 ( 2 τ + B 6, where (6.2 (a,a 2 : ( 0, γ 6,b + γ 6,d s N, (b,b 2 : ( a + cs 2N, γ 6,b γ 6,b + γ 6,d s γ 6,d s N, and the overbars mean (mod and (mod N respectivey. By Proposition.8, t (/,s/n (6τ e πi(b b 2 +b +b 2 b a 2 a ( ( a 2πia 2 q Piecing together the four ead terms and incuding h s, ( ( α s :h s e 2πi 2 γ,b d ( β s ( i 2π ω ds e πi(b b 2 +b +b 2 b a 2 a 2 ( e 2πia 2, so (6.24 P m,s (τ α s (a,b,c,d q v +... ( c/2 e πi 2 {4ac} We consider the expansion of the twisted term. We reca the defintions of a,b,c,d in (6. and note that a a,a 2 a 2. After much cacuation we obtain (6.25 so (6.26 α s α g s ν0 ( ν e 2πi 24 (2 P m,s ( { g (τ α s { γ,b c 4 } ν e πi 2 { 4c2 ν } e πi(b b 2 b b 2 +b b +b 2 b 2 a 2 (b b, ν0 [( ν e 2πi 24 (2 { γ,b c 4 } ν ν ]} e 2πi{ c (b b 2 b b 2 +b b +b 2 b 2 a 2 (b b} q v. To demonstrate that the second exponentia is in fact, we first note that (6.27 b b cν/ and b 2 b 2 + γ 6,b γ 6,b + γ 6,d s γ 6,d s N. Noting that a is odd and 4N c, { e 2πi c2 ν (b b 2 b b 2 +b b +b 2 b 2 a 2 (b b { e 2πi c2 ν 6 + c { }} { ν 2 a 2 e 2πi c2 ν 6 + c 2 } e 2πi { c2 ν {( γ6,b +γ 6,d s N {( a+ cs 2N (b + 2 b 2 +a 2 c ν }} ν }} { c e 2πi 2 ( γ6,b c ν }.

18 8 ERIC MORTENSON To show c 2 (γ 6,b c ν Z it suffices to show γ 6,b c a γ6,b Z. Reca SL c/2 γ 2 (Z, so 6,d a γ 6,d γ 6,b c 2, which impies γ 6,b ( c 2 (mod. This gives that γ 6,b c Z. For exampe, if γ 6,b (mod then c 2 r + 2 which impies that c (mod. It then foows (6.28 P m,s ( (τ ( { g α s α s ( v ν0 ( ν e 2πi 24 (2 q v References { γ,b c 4 } ν } q v +... [AB] G. Andrews and B. Berndt, Ramanujan s Lost Notebook Part I, Springer-Verag, New York, [AO] S. Ahgren and K. Ono, Congruence properties for the partition function, Proc. Nat. Acad. Sci. USA 98 (200, pages [AG] G. Andrews and F. Garvan, Dyson s crank of a partition, Bu. Amer. Math. Soc. 8 (988, pages [AP] G. Andrews and P. Paue, MacMahon s Partition Anaysis XI: Broken diamonds and moduar forms, Acta Arithmetica, accepted for pubication. [AS] A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 4 (954, pages [D] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge 8 (944, pages 0-5. [G] F. Garvan, New combinatoria interpretations of Ramanujan s partition congruences mod 5,7 and, Trans. Amer. Math. Soc. 05 (988, pages [K] N. Kobitz, Introduction to Eiptic Curves and Moduar Forms, Graduate Texts in Mathematics 97, Springer-Verag, New York, 99. [KL] D. Kubert and S. Lang, Moduar Units, Grundehren der mathematischen Wissenschaften 244, Springer- [O] Verag, New York, 98. K. Ono, The Web of Moduarity: arithmetic of the coefficients of moduar forms and q-series, CBMS Regiona Conference Series in Mathematics 02, American Mathematica Society, [O] K. Ono, Distribution of the partition function moduo m, Ann. of Math. 5 (2000, pages [O2] K. Ono, Nonvanishing of quadratic twists of moduar L-functions and appications to eiptic curves, J. Reine Angew. Math. 5 (200, pages [M] K. Mahburg, Partition Congruences and the Andrews-Garvan-Dyson Crank, Proc. Nat. Acad. Sci. USA 02 (2005, pages [Ma] P.A. MacMahon, Combinatory Anaysis, 2 vos., Cambridge University Press, Cambdrige, (Reprinted: Chesea, New York, 960 [R] H. Rademacher, Topics in Anaytic Number Theory, Grundehren der mathematischen Wissenschaften 69, Spring-Verag, New York, 967. [S] J.-P., Serre, Divisibiité de certaines fonctions arithmétiques, L Ens. Math. 22 (976, pages Department of Mathematics, Penn State University, University Park, Pennsyvania 6802 E-mai address: mort@math.psu.edu