THE ANDREWS-STANLEY PARTITION FUNCTION AND p(n): CONGRUENCES

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1 THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES HOLLY SWISHER Abstract R Stanley formulated a partition function tn which counts the number of partitions π for which the number of odd parts of π is congruent to the number of odd parts in the conjugate partition π mod 4 In light of G E Andrews work on this subject, it is natural to ask for relationships between tn and the usual partition function pn In particular, Andrews showed that the mod 5 Ramanujan congruence for pn also holds for tn In this paper we extend his observation by showing that there are infinitely many arithmetic progressions An + B, such that for all n 0, whenever l 5 is prime and j 1 tan + B pan + B 0 mod l j 1 Introduction and statement of results Let n be a nonnegative integer Recall that pn counts the number of partitions π of n, and p0 is defined to be 1 For a partition π, let Oπ be the number of odd parts in π Let π denote the conjugate partition, which is obtained by reading the columns instead of the rows of the Ferrers diagram for π [And98] Stanley s partition function tn counts the number of partitions π of n for which Oπ Oπ mod 4 The generating function for pn is known to equal the following infinite product throughout let q := e πiz, and q h := e πiz h 11 F q := pnq n = n=1 1 1 q n Notice that F q converges absolutely for all z H, the upper half of the complex plane In [And04], Andrews proves that the generating function for tn can also be written as an infinite product 1 Gq := tnq n = F qf q4 5 F q 3 F q F q 16 5 Andrews proved in [And04] that for all n 0, 13 t5n mod 5, showing that the classical mod 5 Ramanujan congruence for pn also holds for tn He did this by proving a certain partition identity using q-series, which he said cries out for a combinatorial proof Yee, Sills, and Boulet have all proven this identity combinatorially, see [Yee04], [Sil04], and [Bou06] In addition, Berkovich Date: August 3, Mathematics Subject Classification Primary 11P8, 11P83 1

2 HOLLY SWISHER and Garvan [BG06] have proven the congruence 13 combinatorially by deriving statistics related to the Andrews-Garvan crank and the 5-core crank which divide t5n + 4 into 5 equinumerous classes This extends the famous work of Garvan, Kim, and Stanton [GKS90] It is natural to ask whether the congruence modulo 5 is just one of many congruences shared by tn and pn In celebrated work of Ono and Ahlgren it has been shown that there are vastly many congruences for pn beyond the Ramanujan ones see [Ono00], [Ahl00], and [AO01] Here we show that the mod 5 congruence tn and pn share is not an isolated example Theorem 11 Let l 5, l 13 be prime, and j a positive integer There are infinitely many non-nested arithmetic progressions An+B such that for all n 0 we have tan + B pan + B 0 mod l j Remark In fact, Theorem 11 follows from a more precise statement see Theorem 34 In Section, we will recall some basic facts about integral and half-integral weight modular forms, the Shimura correspondence, and a theorem of Serre used in the proof of Theorem 11 In Section 3, we will give the proof of Theorem 11 Preliminaries for proof of Theorem 11 First we fix some notation Suppose w 1 Z, N is a positive integer which is divisible by 4 if w Z, and χ is a Dirichlet character mod N Let M w Γ 0 N, χ resp S w Γ 0 N, χ be the usual space of holomorphic modular forms resp cusp forms on the congruence subgroup Γ 0 N, with Nebentypus character χ There are several operators which act on spaces of modular forms The first we will discuss are Hecke operators The results in this section, and more information on modular forms can be found in [Ono04] 1 Hecke operators We will define Hecke operators for both integer and halfinteger weight modular forms We begin with the integer weight case Definition 1 With the notation above, we let fz := anqn M k Γ 0 N, χ where k is an integer, and p N is prime The action of the Hecke operator T p is defined by fz T p := apn + χpp k 1 an/p q n, where an/p = 0 when p n We note that the half-integer weight case is somewhat different Definition Let fz := anqn M λ+ 1 Γ 04N, χ where λ is an integer and p 4N is prime The action of the half-integral Hecke operator T p is defined by

3 THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES 3 fz T p := ap n + χp where an/p = 0 when p n 1 λ These operators are useful due to the following p n p λ 1 an p 1 + χp λ p λ 1 an/p q n, Proposition 3 If fz M k Γ 0 N, χ, where k Z, then fz T p M k Γ 0 N, χ Similarly, if fz M λ+ 1 Γ 04N, χ, where λ Z, then p fz T p M λ+ 1 Γ 04N, χ Furthermore, both T p and T p take cusp forms to cusp forms Other operators We define two other operators U and V which act on formal power series If d is a positive integer, then we define the U-operator Ud by 1 cnq n Ud := cdnq n, n= and the V -operator V d by cnq n V d := n= n= n= cnq dn These two operators also act on spaces of modular forms First we consider the integer weight case Proposition 4 Suppose fz M k Γ 0 N, χ where k Z 1 If d is a positive integer and d N, then If d is any positive integer, then fz Ud M k Γ 0 N, χ fz V d M k Γ 0 Nd, χ Moreover, both Ud and V d take cusp forms to cusp forms The half-integer weight case is slightly more complicated Proposition 5 Suppose fz M λ+ 1 Γ 04N, χ where λ Z 1 If d is a positive integer and d N, then fz Ud M λ+ 1 If d is any positive integer, then fz V d M λ+ 1 Γ 0 4N, Γ 0 4Nd, 4d χ 4d χ

4 4 HOLLY SWISHER Furthermore, again for the half-integral case both Ud and V d take cusp forms to cusp forms 3 Shimura correspondence Here we recall the famous Shimura correspondences [Shi73] which give a means of mapping half-integer weight cusp forms to even integer weight modular forms Definition 6 Let fz = n=1 cnqn S λ+ 1 Γ 04N, χ, with λ 1, and let t be a square-free integer Define the Dirichlet character ψ t by ψ t n = χn 1 λ t n n If we define complex numbers A t n by then n=1 A t n n s := Ls λ + 1, ψ t S t fz := n=1 A t nq n n=1 ctn n s, is a modular form in M λ Γ 0 N, χ Furthermore, if λ then S t fz is a cusp form When λ = 1 there are conditions here omitted which guarantee S t fz is a cusp form From the definition above, it is not hard to show that the Shimura correspondences commute with the Hecke operators in the following way Proposition 7 Let fz S λ+ 1 Γ 04N, χ with λ 1 If t is a square-free integer and p 4Nt is prime, then S t fz T p = S t fz T p 4 A Theorem of Serre First we recall that the following subgroup of Γ 0 N { } a b Γ 1 N := SL c d Z : a d 1 mod N, and c 0 mod N has the property that: 3 M k Γ 1 N = χ M k Γ 0 N, χ 4 S k Γ 1 N = χ S k Γ 0 N, χ The following theorem due to Serre see [Ono00] and [Ser76] is a crucial component to the proof of Theorem 11 The theorem arises from the existence of certain Galois representations with special properties More details can be found in [Ono04], and [Ser76] Theorem 8 Let l 5 be prime, and k Z A positive proportion of the primes p 1 mod N have the property that fz T p 0 mod l for every fz that is the reduction modulo l of a cusp form in S k Γ 1 N Z[[q]]

5 THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES 5 3 Proof of Theorem 11 We will see that using the works of Serre and Shimura, the proof of Theorem 11 boils down to the existence of a half-integral weight cusp form satisfying certain key properties This is given in the following theorem We make the following notation for ease of exposition ln + 1 G l z := t q n n 0 ln 1 mod 4 Theorem 31 Let j 1 be a positive integer and l 5 prime, l 13 For an appropriately chosen positive integer K, there is a cusp form g l,j z S Kl j+1 Kl j 1 1 Γ1 304l having integer coefficients, such that 4 G l z g l,j z mod l j 31 Deduction of Theorem 11 from Theorem 31 Before we prove Theorem 31, we will show how Theorem 11 is proved from it We use techniques of Ono and Ahlgren previously used on pn see [Ono00] and [Ahl00] and extend them to work on tn and pn simultaneously Consider the Shimura lift of g l,j z to an integer weight cusp form with integer coefficients We have that 31 S t g l,j S Kl j+1 Kl j 1 Γ 1 304l Z[[q]] By Theorem 8, there are infinitely many primes p 1 mod 304l such that every reduction mod l j of every form in the space S Kl j+1 Kl j 1 Γ 1 304l Z[[q]] gets annihilated mod l j by the integer weight Hecke operator T p In particular, by equation 31 and Proposition 7, it follows that there are infinitely many primes p 1 mod 304l such that for any square-free integer t Thus in particular, S t g l,j z T p = S t g l,j z T p 0 mod l j 3 g l,j z T p 0 mod l j If we write g l,j z = n=1 cnqn, and let λ l,j = Kl j+1 Kl j 1 /, then Definition and 3 say that n=1 cp n + 1 λ l,j p n p p λ l,j 1 cn + 1 λ l,j p n p λl,j 1 c p q n 0 mod l j Thus for each of the infinitely many primes p for which equation 3 holds, we have 1 cp λ l,j n 1 n + p λl,j 1 λ l,j n cn + p p p p λl,j 1 c p 0 mod l j for all positive integers n Replacing n by np, the middle term vanishes to give us that 1 33 cp 3 λ l,j n n + p p λl,j 1 c 0 mod l j p

6 6 HOLLY SWISHER We restrict our attention further by only considering n which are not divisible by p For these n, cn/p is defined to be 0, so equation 33 becomes cp 3 n 0 mod l j Combining this with Theorem 31 we obtain the following proposition Proposition 3 A positive proportion of the primes p 1 mod 304l have the property that p 3 ln + 1 t 0 mod l j 4 for all n, where ln 1 mod 4 and n, p = 1 The work of Ono and Ahlgren intersects nicely with this analysis of tn Define F l z by ln + 1 F l z := p q n 4 n 0 ln 1 mod 4 In [Ahl00] see Theorem 1, Ahlgren proves the following theorem Theorem 33 Let l 5 be prime, and j 1 a positive integer There exists a cusp form 1 f l,j z S l j l j 1 1 Γ 0 576l, such that f l,j z F l z mod l j Notice that both ηz l /ηl z 1 mod l, and ηz l /ηlz 1 mod l induction we can argue that for any j 1, 34 ηz lj+1 1 mod l j, and ηzl 1 mod l j ηl z lj 1 ηlz lj 1 Define K 1 ηz lj+1 ηz lj 35 fl,j z := f l,j z ηl z lj 1 ηlz lj 1 Then f l,j z f l,j z mod l j Since the second and third factors in 35 are holomorphic modular forms on Γ 0 l, we see that f l,j z S Kl j+1 Kl j 1 1 Γ1 576l We can realize f l,j z as a cusp form on the group Γ 1 304l Serre s Theorem gives a statement about every reduction mod l j of a cusp form in S Kl j+1 Kl j 1 Γ 1 304l Z[[q]], thus we can apply Theorem 8 to S t f l,j z and S t g l,j z simultaneously We conclude the following Theorem 34 Let l 5, l 13 be prime A positive proportion of the primes p 1 mod 304l have the property that p 3 ln + 1 p 3 ln + 1 t p 0 mod l j 4 4 for all positive integers n, where p, n = 1 j By

7 THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES 7 Theorem 11 follows easily from Theorem 34 3 Preliminaries for the proof of Theorem 31 Now it only remains to prove Theorem 31 Recall Dedekind s eta-function 36 ηz := q 1 4 Π n=1 1 q n From equations 11 and 1 we deduce that gz = tnq n 1 4 = ηz η16z 5 ηzη4z 5 η3z For ease of notation, we make the following abbreviations Let δ l be the positive integer and define 1 β l l 1 such that δ l := l 1 4, 4β l 1 mod l Thus we have that ηlz l gz = Π n=11 q ln l tnq n+δ l Using the definitions of the U and V -operators, we obtain the following 37 [ ] ηlzl gz Ul ηz l V 4 = tln + β l q 4n+ 4β l 1 l Letting k = 4n + 4β l 1/l, we see that ln + β l = lk Thus we can show that in fact [ ] ηlzl gz Ul 38 ηz l V 4 = G l z Write E j z = ηzl ηl z It is clear from the definitions of the U and V -operators that they preserve congruences Thus in light of 34, we have that for any K N [ ] ηlzl gz Ul E j z K g l,j z := ηz l V 4 G l z mod l j By properties of eta-quotients and the U, V operators, we see that g l,j z has integer coefficients, and that g l,j z is modular over the group Γ 0 304l with weight Klj+1 Kl j 1 1 We wish to show that g l,j z vanishes at all the cusps of Γ 0 304l

8 8 HOLLY SWISHER 33 Proof of Theorem 31 A complete set of representatives for the cusps of Γ 0 N, where N is a positive integer, is given by [Mar96] 39 { ac c Q : c N, 1 a c N, gcda c, N = 1, a c distinct modulo gcd c, N } c We recall the definition of the slash operator [Kob93] If fz is a function on the a b upper half-plane, λ 1 Z, and GL + c d R, then a b 310 fz λ := ad bc λ cz + d λ az + b f c d cz + d Moreover, let γ a be the matrix in SL c Z that takes to a c We know from [Mar96] that the expansion of a modular form fz of weight λ R at the cusp a c is of the form fz λ γ a = k qα + c for some nonzero constant k and α Q Thus α is the order of vanishing of fz at the cusp a c We are interested in the expansion of g l,jz at the cusps of Γ 0 3l However, V 4 cannot introduce poles so it suffices to consider the Laurent series of [ ηlzl gz Ul E j z K ] ηz l It is a fact [Mar96] that for all γ SL Z, ηz l l γ = k q l 4 + Since ηlz l gz Ul E j z K is modular on Γ 0 3l, all that remains to be shown is that ηlz l gz Ul E j z K has order of vanishing > l/4 at all cusps of Γ 0 3l It is easy to compute [Mar96] that 0 if l c, ord a E c jz K = Kl j 1 δ l if l c, Kl j 3 l + 1δ l if l c So we choose K large enough such that it kills all poles of ηlz l gz Ul at cusps a c with l c For convenience, let hz := ηlz l gz Also, let a b A := cl SL d Z correspond to the cusp a/cl of Γ 0 304l We will use the following proposition which can be obtained easily from the definition of the U-operator Proposition 35 If l is a prime positive integer and P z = n 1 cnqn, then P z Ul = 1 l 1 z + j P l l j=0

9 THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES 9 By Proposition 35, l 1 hz Ul l 1 A = l l 5 4 hz l 1 v=0 1 v 0 l Define for each v, an integer w v such that 304 w v, and Also, define w v a + vcl 1 b + vd mod l a + vcl b+vd aw v w vvcl γ v := l cl 3 d w v cl A calculation shows that γ v SL Z and that 1 v 1 wv 311 A = γ 0 l v 0 l Using 311, we can obtain the following l 1 31 hz Ul l 1 A = l l wv hz l 1 γ 0 0 l v=0 A We note that this type of argument can be found in greater generality in [Tre06] We can calculate hz l 1 γ 0 directly see [Mar96] to obtain hz l 1 γ 0 = a c nqh n c, n n c where and δ l h c n c = δ l h c 1 δ l h c + Plugging back into 31 we get that for c=1,,8,3, for c=4, for c=16, 3 if c = 1 8 if c = h c = if c = 4 1 if c = 8, 16, 3 hz Ul l 1 A = l 1 a c ne πizn hcl n n c l 1 v=0 e πiwv n hcl One can show that w v runs through the residue classes mod l as v does Since 304 w v we have h c w v for all c Since also h c, l = 1, then in fact w v /h c runs through the residue classes mod l as v does Hence, 313 hz Ul l 1 A = l 1 n l a c nq n n c h c l 1 v=0 e πivn l = alnqh n c n nc l

10 10 HOLLY SWISHER We must work through each case to show that if hz Ul l 1 A = q α +, then α > l 4 Since the calculations for the cases are very similar, we will only show the details of the trickiest case here, when c = 4 When c = 4, 313 yields hz Ul l 1 A = n δ l 1 l a c lnq n = q α + Thus α = m, where m is an integer satisfying m δ l 1/l Then choose the least integer x 0 such that lm = δ l 1 + x = l 6 + 4x 0 mod l 4 Note that if we don t allow l = 13, then we must have x in order for the congruence to hold In this case, α = 1 l x > l 4l 4 The other cases are similar, but don t require the exclusion of any primes Remark In [Tre06], Treneer proves general results such as equation 31 by relating eta-quotients and their images under the U-operator to cusp forms References [Ahl00] Scott Ahlgren Distribution of the partition function modulo composite integers M Math Ann, 3184: , 000 [And98] George E Andrews The theory of partitions Cambridge Mathematical Library Cambridge University Press, Cambridge, 1998 Reprint of the 1976 original [And04] George E Andrews On a partition function of Richard Stanley Electronic J Combin, 11, 004 [AO01] Scott Ahlgren and Ken Ono Congruence properties for the partition function Proc Natl Acad Sci USA, 983: electronic, 001 [BG06] Alexander Berkovich and Frank G Garvan On the Andrews-Stanley refinement of Ramanujan s partition congruence modulo 5 and generalizations Trans Amer Math Soc, 358: electronic, 006 [Bou06] Cilanne E Boulet A four-parameter partition identity Ramanujan J, 13:315 30, 006 [GKS90] Frank Garvan, Dongsu Kim, and Dennis Stanton Cranks and t-cores Invent Math, 1011:1 17, 1990 [Kob93] Neal Koblitz Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics Springer-Verlag, New York, second edition, 1993 [Mar96] Yves Martin Multiplicative η-quotients Trans Amer Math Soc, 3481: , 1996 [Ono00] Ken Ono Distribution of the partition function modulo m Ann of Math, 1511:93 307, 000 [Ono04] Ken Ono The web of modularity: arithmetic of the coefficients of modular forms and q-series, volume 10 of CBMS Regional Conference Series in Mathematics Published for the Conference Board of the Mathematical Sciences, Washington, DC, 004 [Ser76] Jean-Pierre Serre Divisibilité de certaines fonctions arithmétiques Enseignement Math, 3-4:7 60, 1976 [Shi73] Goro Shimura On modular forms of half integral weight Ann of Math, 97: , 1973 [Sil04] Andrew V Sills A combinatorial proof of a partition identity of Andrews and Stanley Int J Math Math Sci, 45-48: , 004 [Sta0] R P Stanley Problem Amer Math Monthly, 109:760, 00

11 THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES 11 [Sta05] Richard P Stanley Some remarks on sign-balanced and maj-balanced posets Adv in Appl Math, 344:880 90, 005 [Tre06] Stephanie Treneer Congruences for the coefficients of weakly holomorphic modular forms Proc London Math Soc 3, 93:304 34, 006 [Yee04] Ae Ja Yee On partition functions of Andrews and Stanley J Combin Theory Ser A, 107:313 31, 004 Department of Mathematics, Oregon State University, Corvallis, OR address: swisherh@mathoregonstateedu