Congruences of Multipartition Functions Modulo Powers of Primes. Tianjin University, Tianjin , P. R. China

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1 Congruences of Multipartition Functions Modulo Powers of Primes William Y.C. Chen 1, Daniel K. Du, Qing-Hu Hou 3 and Lisa H. Sun 4 1,3 Center for Applied Mathematics Tianjin University, Tianjin 30007, P. R. China,4 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin , P. R. China 1 chenyc@tju.edu.cn, dukang@mail.nankai.edu.cn, 3 hou@nankai.edu.cn, 4 sunhui@nankai.edu.cn Abstract. Let p r n denote the number of r-component multipartitions of n, and let S γ,λ be the space spanned by ηz γ φz, where ηz is the Dedekind s eta function and φz is a holomorphic modular form in M λ SL Z. Inthispaper,weshowthatthegeneratingfunctionofp r mk n+r with respect to n is congruent to a function in the space S γ,λ modulo m k. As special cases, this relation leads to many well known congruences including the Ramanujan congruences of pn modulo 5, 7, 11 and Gandhi s congruences of p n modulo 5 and p 8 n modulo 11. Furthermore, using the invariance property of S γ,λ under the Hecke operator T l, we obtain two classes of congruences pertaining to the m k -adic property of p r n. AMS Classification. 05A17, 11F33, 11P83 Keywords. modular form, partition, multipartition, Ramanujan-type congruence 1 Introduction The objective of this paper is to use the theory of modular forms to derive certain congruences of multipartitions modulo powers of primes. Recall that an ordinary partition λ of a nonnegative integer n is a nonincreasing sequence of positive integers whose sum is n, where n is called the weight of λ. The partition function pn is defined to be the number of partitions of n. A multipartition of n with r components, as called by Andrews [], also referred to as an r-colored partition, see, for example [9,11] is an r-tuple λ = λ 1,...,λ r of partitions whose weights sum to n. The number of r-component multipartitions of n is denoted by p r n. Multipartitions arise in combinatorics, representation theory, and physics. As pointed out by Fayers [1], the representations of the Ariki Koike algebra are naturally indexed by multiparititions. Bouwknegt [8] showed that the Durfee square formulas of multipartitions are useful in deriving expressions 1

2 for the characters of modules of affine Lie algebras in terms of the universal chiral partition functions. For the partition function pn, Ramanujan [5 8] proved that pan+b 0 mod M, 1.1 for all nonnegative integers n and for A, B, M = 5, 4, 5, 7, 5, 7 and 11, 6, 11. In general, congruences of form 1.1 are called Ramanujan-type congruences. For m = 5 and 7, Watson [31] proved that pm k n+β m,k 0 mod m r k, 1. where k 1, β m,k 1/ mod m k, r k = k for m = 5 and r k = k/ +1 for m = 7. The case m = 5 in 1. was considered by Ramanujan, see Berndt and Ono [7]. Atkin [3] showed that 1. is also valid for m = 11. When M is not a power of 5,7 or 11, Atkin and O Brien [5] discovered the following congruence p n+37 0 mod 13. Using the theory of modular forms, Ono [3] proved that, for any prime m 5 and positive integer k, there is a positive proportion of primes l such that m k l 3 n+1 p 0 mod m 1.3 holds for every nonnegative integer n coprime to l. Weaver [3] gave an algorithm for finding the values of l in 1.3 for primes 13 m 31. Recently, Folsom, Kent, and Ono [13] provided a very general theorem which gives new generalized partition congruences systematically. In this framework, they proved that if 5 m 31 is a prime and k is a positive integer, then there exists an integer A m b 1,b,k such that m b 1 n+1 m b n+1 p A m b 1,b,kp mod m k 1.4 for all positive integers n and b 1 b mod larger than some fixed integer. Ramanujan-type congruences of p r n have been extensively studied, see, for example [, 4, 14, 15, 17, 19,, 30]. Gandhi [14] derived the following congruences of p r n by applying the identities of Euler and Jacobi p 5n+3 0 mod 5, 1.5 p 8 11n+4 0 mod With the aid of Sturm s theorem [9], Eichhorn and Ono [11] computed an upper bound CA,B,r,m k such that p r An+B 0 mod m k

3 holdsforallnonnegativeintegersnifandonlyifitistrueforn CA,B,r,m k. For example, to prove 1.5, it suffices to check that it holds for n 3. In the same vain, one can prove 1.6 by verifying that it holds for n 11. Treneer [30] extended 1.3 to weakly holomorphic modular forms and showed that, for any prime m 5 and positive integers k, there is a positive proportion of primes l such that p r m k l μr n+r 0 mod m foreverynonnegativeinteger ncoprimetol, whereμ r equals1ifr isevenand 3 if r is odd. Using the methods of Folsom, Kent, and Ono [13], Belmont et al. [6, Corollary 1.] generalized congruence 1.4 to the cases of p r n. They proved that if the rank of the corresponding space is no more than 1, then there exists an integer C l r,b 1,b,k such that p r m b 1 +r C l r,b 1,b,kp r m b +r mod m k, 1.7 where n is a positive integer and b 1 b mod are large enough integers. The aim of this paper is to study congruence properties of p r n modulo powers of primes. For example, we shall derive the following two classes of congruences m k l μk 1 n+r p r 0 mod m k, 1.8 m k l i n+r m k l M+i n+r p r p r mod m k, 1.9 where r is an odd integer, l is any prime other than,3 and m, and μ is an arbitrary positive integer, K and M are fixed positive integers, and n is a positive integer coprime to l. To derive congruences of p r n, one may consider the congruence properties of the generating functions of p r n. For the case of ordinary partitions, i.e., r = 1, Chua [10] showed that mn+1 p q n ηz γm φ m z mod m, 1.10 mn 1 mod where ηz is Dedekind s eta function, γ m is an integer depending on m, and φ m z is a holomorphic modular form. Ahlgren and Boylan [1] extended 1.10 to congruences modulo powers of primes, namely, m k n+1 F m,k z = p q n ηz γ m,k φ m,k z mod m k, m k n 1 mod

4 where γ m,k is an integer and φ m,k z is a holomorphic modular form. In order to prove the existence of congruences of p r n modulo powers of primes, Brown and Li [9] introduced the generating function G m,k,r z q n mod m k, 1.1 m= n r m p r n+r and showed that G m,k,r z is a modular form of level 576m 3. Kilbourn [18] used the generating function mn+r H m,k,r z p r q n mod m k, 1.13 mn r mod and proved that H m,k,r z is a modular form of level 576m. However, due to the large dimensions of the spaces M λ Γ 0 576m 3 and M λ Γ 0 576m, it does not seem to be a feasible task to compute explicit bases. In other words, to derive explicit congruence formulas of p r n, it is desirable to find a generating function of p r n that can be expressed in terms of modular forms of a small level. In this paper, we find the following extension of the generating function F m,k z, namely, m k n+r F m,k,r z = p r q n, 1.14 m k n r mod whereq = e πiz. WeshowthatF m,k,r ziscongruenttoaweaklyholomorphic function modulo m k. More precisely, we find F m,k,r z ηz γ m,k,r φ m,k,r z mod m k, 1.15 where γ m,k,r is an integer and φ m,k,r z is a holomorphic modular form in M λm,k,r SL Z. Noting that any element of M λm,k,r SL Z can be expressed as a polynomial of the Eisenstein series E 4 z and E 6 z, this enables us to derive explicit congruences of the generating function of p r n modulo m k. If φ m,k,r z = 0, then 1.15 yields a Ramanujan-type congruence as follows m k n+r p r 0 mod m k For example, it is easily checked that φ 5,1, z = 0 and φ 11,1, z = 0, hence Gandhi s congruences 1.5 and 1.6 are the consequences of We also find p 5 n+3 0 mod 5,

5 p 8 11 n+81 0 mod 11, 1.18 since φ 5,, z = 0 and φ 11,,8 z = 0. For more congruences of form 1.16, see Table 5.. On the other hand, if φ m,k,r z = 0 in 1.15, we may use Yang s method [33] to find congruences of form 1.8. For example, since F 5,,3 z is congruent to a modular form in the invariant space S 1,48 of T 5 modulo 5, we have Preliminaries p n+3 0 mod 5. To make this paper self-contained, we recall some definitions and facts on modular forms. In particular, we shall use the U-operator, the V-operator, the Hecke operator, and the twist operator on modular forms. Let k 1 Z be an integer or a half-integer, N be a positive integer with 4 N if k Z and χ be a Nebentypus character. We use M k Γ 0 N,χ to denote the space of holomorphic modular forms on Γ 0 N of weight k and character χ. The corresponding space of cusp forms is denoted by S k Γ 0 N,χ. If χ is the trivial character, we shall write M k Γ 0 N and S k Γ 0 N for M k Γ 0 N,χ and S k Γ 0 N,χ, respectively. Moreover, we write SL Z for Γ 0 1. Let fz M k Γ 0 N,χ with the following Fourier expansion at fz = n 0anq n, where q = e πiz. Let us recall some operators acting on fz. Let γ = a b c d be a real matrix with positive determinant. The k slash operator k is defined by f k γz = detγ k/ cz +d k fγz,.1 where γz = az +b cz +d. In particular, let l be an integer and 0 1 γ l = l 0 5.

6 The Fricke involution W l is given by f W l = f k γ l.. The U-operator U l and V-operator V l are defined by fz U l = n 0alnq n.3 and fz V l = n 0anq ln..4 It is known that l 1 fz k U l = l k 1 1 μ fz k 0 l μ=0..5 Let ψ be a Dirichlet character. The ψ-twist of fz is defined by f ψz = n 0ψnanq n. Let l be a prime and fz M λ+ 1Γ 0 N,χ be a modular form of halfintegral weight. The Hecke operator T l is defined by fz T l = n 0 al n+χl 1 λ n l l λ 1 an+χl l λ 1 a n l q n..6 We will use the following level reduction properties of the operators U l and Tr l = U l +l k 1 W l see [0, Lemma 1] and [10, Lemma.]. Lemma.1 Let k Z, N be a positive integer, χ be a character modulo N, and fz M k Γ 0 N,χ. Assume that l is a prime factor of N and χ is also a character modulo N/l. 1. If l N, then f U l M k Γ 0 N/l,χ.. If N = l and χ is the trivial character, then f Tr l M k SL Z. In the proof of congruence 1.15 on the generating function F m,k,r z, we need the following relation ηγz = ε a,b,c,d cz +d 1 ηz,.7 6

7 a b where γ = SL c d Z, ε a,b,c,d is a th root of unity, and ηz is Dedekind s eta function as given by As a special case, we have ηz = q 1 1 q n..8 n=1 η 1/z = z/i ηz..9 3 The generating function of p r n modulo m k In this section, we derive the congruence of the generating function F m,k,r z defined by 1.14, namely, m k n+r F m,k,r z = p r q n. m k n r mod Theorem 3.1 Let m 5 be a prime, and let k and r be positive integers. Then there exists a modular form φ m,k,r z M λm,k,r SL Z such that F m,k,r z ηz γ m,k,r φ m,k,r z mod m k, 3.1 where λ m,k,r = { m k m k 1 r γ m,k,r+r if k is odd, m k m k 1 r γ m,k,r+r if k is even, 3. γ m,k,r = β m,k,r r m k, 3.3 and β m,k,r is the unique integer in the range 0 β m,k,r < m k congruent to r/ modulo m k. The first step of the proof of Theorem 3.1 is to express F m,k,r z in terms of a modular form. Consider the η-quotient r ηm k z mk f m,k,r z =, 3.4 ηz which is a modular form in M m k 1r Γ 0 m k, kr. The following lemma m shows that F m,k,r z can be obtained from f m,k,r z by applying the Uoperator and the V-operator. 7

8 Lemma 3. Let m 5 be a prime, and let k and r be positive integers. Then we have F m,k,r z = f m,k,rz U m k V. 3.5 ηz mk r Proof. Since we find that p r nq n = n=0 f m,k,r z = q mk 1 r = q mk 1 r n=1 n=1 1 1 q n r, 1 1 q n 1 q mkn mk r r n=1 p r nq n 1 q mkn mkr. n=0 n=1 Applying the operator U m k to the above relation, we obtain f m,k,r z U m k = q mk 1 m k r n=0 p r m k nq n 1 q n mkr. 3.6 Let0 β m,k,r m k 1betheintegeruniquelydetermined bythecongruence β m,k,r r mod m k. Substituting n by n + β m,k,r in the summation in m k 3.6, we find f m,k,r z U m k = n=0 n=1 k 1+βm,k,r p r m k n+β m,k,r q n+rm m k which belongs to Z[[q]]. So we deduce that n=0 k 1+βm,k,r p r m k n+β m,k,r q n+rm m k Applying the operator V, we get 1 q n mkr, n=1 = f m,k,rz U m k n=1 1 qn. mk r p r m k n+β m,k,r q n+β m,k,r r m k = f m,k,rz U m k V. 3.7 ηz mk r n=0 Substituting n + β m,k,r r by n in 3.7, the sum on the left-hand side m k becomes F m,k,r z. This completes the proof. The second step of the proof of Theorem 3.1 is to derive a congruence relation for f m,k,r z U m k modulo m k. 8

9 Theorem 3.3 Let m 5 be a prime, and let k and r be positive integers. Then there exists a modular form G m,k,r z M wm,k,r SL Z such that f m,k,r z U m k G m,k,r z mod m k, where w m,k,r = { m k m k 1 1 r if k is odd, 3m k m k 1 1 r if k is even. Proof. Let where ηz m g m,k,r z = ηmz ck m k 1 r 1 if k is odd, c k = if k is even. Since g m,k,r z is an η-quotient, using the modular transformation property due to Gordon, Hughes, and Newman [16, 1], see also, [, Theorem 1.64], we deduce that g m,k,r z M ck m k m k 1 r kr Γ 0 m,. m Moreover, since 1 q n m 1 q mn mod m, we see that which implies that Since f m,k,r z S m k 1r we obtain that ηz m ηmz mod m, g m,k,r z 1 mod m k. 3.8 Γ 0 m k, kr, using Lemma.1 repeatedly, m f m,k,r z U m k 1 S m k 1r kr Γ 0 m,. m Thus, f m,k,r z U m k 1 g m,k,r z is a modular form on Γ 0 m of the trivial character and of weight, w m,k,r = c km k m k 1 r Invoking Lemma.1, we find that + mk 1r. G m,k,r z = f m,k,r z U m k 1 g m,k,r z Tr m 3.9 9

10 is a modular form in M wm,k,r SL Z. where To complete the proof of Theorem 3.3, it remains to show that f m,k,r z U m k 1 g m,k,r z Tr m f m,k,r z U m k mod m k, 3.10 Tr m = U m +m w m,k,r 1 W m, and the operator W m is given by.. By congruence 3.8, we see that the left-hand side of 3.10 equals f m,k,r z U m k +m w m,k,r 1 f m,k,r z U m k 1 g m,k,r z W m mod m k. To prove 3.10, it suffices to show that m w m,k,r 1 f m,k,r z U m k 1 g m,k,r z W m 0 mod m k We only consider the case when k is odd. The case when k is even can be dealt with in the same manner. In light of the transformation formula.9 of the eta function, we find that g m,k,r z W m = m mk m k 1 r 4 mz mk m k 1 r g m,k,r 1 mz = m mk m k 1 r 4 z mk m k 1 r mz/iηmz mk 1r m z/iηz = m m+1mk 1 r 4 i m 1mk 1 r ηmz m ηz m k 1 r Therefore, 3.11 can be deduced from the following congruence m 3mk 1r 4 1 f m,k,r z U m k 1 W m 0 mod m k. 3.1 By the property of the U-operator as in.5, we have m 3mk 1r 4 1 f m,k,r z U m k 1 W m = m k+mk r r+4k 4 = m k+mk r r+4k 4 m k 1 1 μ=0 m k 1 1 μ=0 f m,k,r z m k 1r f m,k,r z m k 1r 10 1 μ 0 m k 1 μm 1 m k 0. Wm. 3.13

11 Using the transformation formula.9 of the eta function, 3.13 can be written as m k 1 1 r m mk r k z mk 1r ηmμ 1 z mk η mμz 1 m k z μ=0 m k 1 1 = m mk r k z r ηz m k r where α μ is a th root of unity. μ=0 α μ η mμz r, m k z For μ = 0, we write μ = m s t where m t. For μ = 0, we set s = k 1 and t = 0. Ineither case, thereexist integersbanddsuchthatbt+dm k s 1 = 1. It follows that mμ 1 m k 0 = t d m k s 1 b m s+1 b 0 m k s 1 Applying the corresponding slash operator to ηz, we obtain that mμz 1 η = ε m k μ m s+1 1 m s+1 z +b z η, z m k s 1 whereε μ isathrootofunity. Sincethecoefficients ofthefourierexpansion ofηzat areintegersandthecoefficient ofthetermwiththelowest degree is 1, the Fourier coefficients of each term in3.14 aredivisible by m mk s 1 r k in the ring Z[ζ ]. Clearly, 0 s k 1. Thus we have m k s 1 r k mk k r k mk k k k for m 5 and k 1. Hence the Fourier coefficients of each term in 3.14 are divisible by m k. So we arrive at 3.1. This completes the proof. We are now in a position to finish the proof of Theorem 3.1. Proofof Theorem3.1. ByTheorem3.3,thereexistsamodularformG m,k,r z M wm,k,r SL Z such that Let. f m,k,r z U m k G m,k,r z mod m k φ m,k,r z = G m,k,rz Δz mk r+γ m,k,r where Δz = ηz is Ramanujan s Δ-function. In the proof of Lemma 3., we have shown that k 1+βm,k,r f m,k,r z U m k = p r m k n+β m,k,r q n+rm m k 1 q n mkr, n=0 11, n=1

12 which implies that the order of the Fourier expansion of f m,k,r z U m k at is at least rm k 1+β m,k,r = mk r+γ m,k,r. m k Thus φ m,k,r z is a modular form in M λm,k,r SL Z. Combining 3.15 and Lemma 3., we conclude that Δz m k r+γm,k,r φ m,k,r z V F m,k,r z ηz mk r as required. = ηz γ m,k,r φ m,k,r z mod m k, 4 Congruences of p r n modulo m k In this section, we apply Theorem 3.1 on the congruence relation for the generating function F m,r,k z and Yang s method [33] to derive two classes of congruences of p r n modulo m k. Let S γ,λ = {ηz γ φz: φz M λ SL Z}. Yang [33] showed that when γ is anoddinteger such that 0 < γ < and λ is anonnegativeeveninteger, S γ,λ isaninvariantsubspaceofs λ+γ/ Γ 0 576,χ 1 under the action of the Hecke algebra. More precisely, for all primes l =,3 and all f S γ,λ, we have f T l S γ,λ. By the invariant property of S γ,λ, we obtain two classes of congruences of p r n modulo m k. Theorem 4.1 Let m 5 be a prime, k be a positive integer, r be an odd positive integer less than m k, and l be a prime different from,3, and m. Then there exists an explicitly computable positive integer K such that p r m k l μk 1 n+r 0 mod m k 4.1 for all positive integers μ and all positive integers n relatively prime to l. There is also a positive integer M such that m k l i n+r m k l M+i n+r p r p r mod m k 4. for all nonnegative integers i and n. 1

13 Proof.Accordingtocongruencerelation3.1,thegeneratingfunctionF m,k,r z is congruent to a modular form in S γm,k,r,λ m,k,r Z[[q]], where λ m,k,r and γ m,k,r are integers as given in 3. and 3.3. It is known that S γm,k,r,λ m,k,r Z[[q]] has a basis {f 1 z,...,f d z} of the form f i z = E 4 z u i E 6 z v i Δz w i, where u i,v i and w i are nonnegative integers satisfying 4u i + 6v i + 1w i = λ m,k,r +γ m,k,r / for more details, see []. Suppose that f i z = n 0a i nq n, where i = 1,,...,d. To prove 4.1, it suffices to show that there exists a positive integer K such that for any 1 i d, for all n coprime to l. a i l μk 1 n 0 mod m k 4.3 From the relation γ m,k,r m k = β m,k,r r, one sees that γ m,k,r and r have the same parity. Since r < m k is odd, we have 0 < γ m,k,r <, and hence S γm,k,r,λ m,k,r is invariant under the Hecke operator T l. So there exists a d d matrix A such that Let X = f 1. f d T l = A A f 1. f d. 4.4 I d l γ m,k,r+λ m,k,r I d 0 Using the property of the basis {f 1 z,...,f d z} under the action of the U-operator as given by Yang [33, Corollary 3.4], we obtain f 1 f 1 g 1 f 1. U s l = A s. +B s. +C s. V l, 4.5 f d f d g d f d where s is a positive integer, g i = f i l, and As,B s, and C s are d d matrices given by As A s 1 = Id 0 X s, 4.6 B s = l λ m,k,r+γ m,k,r 3/ 1 γ m,k,r 1/ 1 A s 1, l 13.

14 C s = l γ m,k,r+λ m,k,r A s 1. Since gcdm,l = 1, the matrix X mod m k is invertible in the ring M consisting of d d matrices over Z m k. By the finiteness of M, we see that there exist integers a > b such that X a and X b are linearly dependent over Z m k, i.e., there exists a constant c Z m k such that X a cx b mod m k. Thus X K ci d mod m k, where K = a b. In view of the relation AμK A μk 1 c μ I d 0 mod m k, we find that A μk 1 0 mod m k. Hence, from 4.5 it follows that f 1 g 1 f 1. U μk 1 l B μk 1. +C μk 1. V l mod m k. f d g d f d Applying the U-operator U l and observing that Ul g i U l = f i = 0, l relation 4.5 leads to the following congruence f 1 f 1. U μk 1 l U l C μk 1. V l mod m k, f d f d namely, which implies 4.3. n 0a i l μk 1 nq n n 0 a i nq ln mod m k, We now turn to the proof of congruence 4.. By the finiteness of M, we see that there exists a positive integer M such that X M I d mod m k. Thus matrix equation 4.6 reduces to the following congruence AM A M 1 Id 0 mod m k. It follows that A M I d mod m k and B M C M 0 mod m k. Thus, relation 4.5 implies f 1 f 1. U M l. mod m k. f d f d So the coefficient of q n is congruent to the coefficient of q lmn in f i z modulo m k forall iandn. Since F m,k,r z is a linear combination off i z with integer coefficients, we obtain congruence 4.. This completes the proof. 14

15 5 Examples In this section, we present some consequences of Theorem 3.1 and Theorem 4.1. We first give some examples for the congruences of the generating function F m,k,r z of p r n. Example 5.1 By Theorem 3.1, we find F m,k,r z ηz γ m,k,r φ m,k,r z mod m k, where γ m,k,r is an integer, φ m,k,r z is a polynomialof Δz and the Eisenstein series E 4 z and E 6 z. Table 5.1 gives the list of explicit expressions of ηz γ m,1,r φ m,1,r z for m 19 and r 7. r m ηz γ m,1,r φ m,1,r z ηz ηz E 4 z 13 8ηz 17 5ηz 14 E 4 z 19 ηz 10 14E 4 z 3 +1Δz 3 5 4ηz 9 7 3ηz 3 E 6 z ηz 9 4E 4 z 3 +6Δz ηz 15 E 6 z 3 +3E 6 zδz 4 5 4ηz 4 E 4 z ηz 4 3E 4 z 4 +8E 4 zδz 13 ηz 0 7E 4 z 3 +4Δz 17 ηz 4 6E 4 z 7 +11E 4 z 4 Δz+4E 4 zδz 19 ηz 0 16E 4 z 6 +18E 4 z 3 Δz+Δz 5 5 ηz 1 E 4 z 7 ηz 13 E 6 z ηz 7 8E 4 z 6 +11E 4 z 3 Δz+5Δz 17 ηz 11 16E 4 z 8 +16E 4 z 5 Δz+4E 4 z Δz 19 ηz5e 6 z 7 +15E 6 z 5 Δz+16E 6 z 3 Δz ηz 6 6E 4 z 3 +6Δz 11 ηz 6 10E 4 z 6 +E 4 z 3 Δz 13 ηz 18 7E 4 z 6 +8E 4 z 3 Δz+6Δz 15

16 17 ηz 18 3E 4 z 9 +3E 4 z 6 Δz+5E 4 z 3 Δz 19 ηz 6 6E 4 z 1 +E 4 z 9 Δz+14Δz ηz 1 E 6 z ηz 5 10E 4 z 9 +6E 4 z 6 Δz+9E 4 z 3 Δz +11Δz 3 17 ηz7e 4 z 13 +E 4 z 10 Δz+E 4 z 7 Δz +3E 4 z 4 Δz Table 5.1: Explicit congruences derived from Theorem 3.1. Example 5. Let 0 β < m k be an integer with β r/ mod m k. If φ m,k,r z 0 mod m k, using Theorem 3.1, we obtain the following Ramanujan-type congruences of multipartition functions p r m k n+β 0 mod m k. 5.1 The values of m and β for r 9 and k = 1, are given in Table 5.. r m,β m,β 1 5,4,7,5,11,6 5,,49,47,11,116 5,3 5,3 3 11,7,17,15 11, ,6 49, ,8,3,5 11,96 6 5,4 5,19 7 5,3,11,9,19,9 5,18,11,86 8 7,5,11,4 11, ,11,19,17,3,9 Table 5.: Ramanujan-type congruences of multipartitions. It can be seen that Table 5. contains the Ramanujan congruences 1.1 of pn modulo 5,7 and 11, as well as Gandhi s congruences 1.5 for p n and 1.6 for p 8 n. The congruences marked by in the table seem to be new. The following examples demonstrate how to derive certain congruences of p r n with the aid of Theorem 4.1. Example 5.3 For the values of l and K l as given in Table 5.3, we have 7 l μk l 1 n+3 p 3 0 mod 7 5. for all positive integers μ and all positive integers n not divisible by l. 16

17 l a l K l Table 5.3: Eigenvalues a l of F 7,1,3 z acted by T l and the corresponding K l. Proof. By Theorem 3.1, we find F 7,1,3 z 3ηz 3 E 6 z mod 7. Since ηz 3 E 6 z belongs to the 1-dimensional space S 3,6, for any prime l =,3,7, there exists an integer a l such that F 7,1,3 z T l a l F 7,1,3 z mod 7. Inspecting the proof of Theorem 4.1, we obtain the corresponding orders K l for which congruence 5. holds. Example 5.4 We have p n+3 0 mod 5 for all integers n coprime to 13 and 5 13 i n i n+3 p 3 p 3 for all nonnegative integers n and i. mod 5 Proof. By Theorem 3.1, F 5,,3 z is congruent to a modular form in the space S 1,48 of dimension 5. Setting f i = ηz 1 E 4 z 35 i Δz i 1 for 1 i 5. Clearly, f 1,f,...,f 5 form a Z-basis of S 1,48 Z[[q]]. Let A be the matrix of T l with respect to this basis. By computing the first five Fourier coefficients of f i and f i T 13 and equating the Fourier coefficients of both sides of 4.4, we find A mod 5, with the corresponding orders K = M = 100. Setting μ = 1 in Theorem 4.1, we complete the proof. Below are two more examples for p 3 n and p 5 n modulo 7. The proofs are analogous to the proof of the above example, and hence are omitted. 17

18 Example 5.5 We have p n+3 0 mod 7 for all positive integers n coprime to 7 and 7 11 i n i n+3 p 3 p 3 mod 7 for all nonnegative integers n and i. Example 5.6 We have p n+5 0 mod 7 for all positive integers n coprime to 17 and 7 17 i n i n+5 p 5 p 5 mod 7 for all nonnegative integers n and i. Acknowledgments. We wish to thank the referee for helpful suggestions. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, and the National Science Foundation of China. References [1] S. Ahlgren and M. Boylan, Arithmetic properties of the partition function, Invent. Math [] G.E. Andrews, A survey of multipartitions: congruences and identities, In: Surveys in Number Theory, K. Alladi, ed., Developments in Mathematics, Vol. 17, Springer, Berlin, 008, pp [3] A.O.L. Atkin, Proof of a conjecture of Ramanujan, Glasg. Math. J [4] A.O.L. Atkin, Ramanujan congruences for p k n, Canad. J. Math [5] A.O.L. Atkin and J.N. O Brien, Some properties of pn and cn modulo powers of 13, Trans. Amer. Math. Soc

19 [6] E. Belmont, H. Lee, A. Musat and S.Trebat-Leder, l-adic properties of partition functions, Monatsh. Math., to appear. [7] B.C. Berndt and K. Ono, Ramanujan s unpublished manuscript on the partition and tau functions with proofs and commentary, Sém. Lothar. Combin [8] P. Bouwknegt, Multipartitions, generalized Durfee squares and affine Lie algebra characters, J. Austral. Math. Soc [9] J.L. Brown and Y.K. Li, Distribution of powers of the partition function modulo l j, J. Number Theory [10] K.S. Chua, Explicit congruences for the partition function modulo every prime, Arch. Math [11] D. Eichhorn and K. Ono, Congruences for partition functions, In: Analytic Number Theory, Vol. 1, Progr. Math. 138, Birkhauser, 1996, pp [1] M. Fayers, Weights of multipartitions and representations of Ariki Koike algebras, Adv. Math [13] A. Folsom, Z.A. Kent and K. Ono, l-adic properties of the partition function, Advances in Math. 9 01, [14] J.M. Gandhi, Congruences forp r nandramanujan sτ function, Amer. Math. Monthly [15] B. Gordon, Ramanujan congruence for p k mod 11 r, Glasgow Math. J [16] B. Gordon and K. Hughes, Multiplicative properties of eta-products II, Cont. Math [17] H. Gupta, Selected Topics in Number Theory, Abacus Press, Turnbridge Wells, [18] T.P. Kilbourn, Congruence properties of Fourier coefficients of modular forms, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 007. [19] I. Kiming and J.B. Olsson, Congruences like Ramanujan s for powers of the partition function, Arch. Math. Basel [0] W. Li, Newforms and functional equations, Math. Ann

20 [1] M. Newman, Construction and application of a class of modular functions, Proc. London Math. Soc. s [] M. Newman, Some theorems about p r n, Canad. J. Math [3] K. Ono, Distribution of the partition function modulo m, Ann. Math [] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, CBMS 10, Amer. Math. Soc., Providence, 004. [5] S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc [6] S. Ramanujan, Some properties of pn, the number of partitions of n, Proc. Cambridge Philos. Soc [7] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. III. Ser xix. [8] S. Ramanujan, Congruence properties of partitions, Math. Z [9] J. Sturm, On the congruence of modular forms, In: Lect. Notes Math , Springer Verlag, pp [30] S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc [31] G.N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math [3] R. Weaver, New congruences for the partition function, Ramanujan J [33] Y. Yang, Congruences of the partition function, Internat. Math. Res. Notices