Rank and crank moments for overpartitions
aa r X i v : . [ m a t h . N T ] J u l RANK AND CRANK MOMENTS FOR OVERPARTITIONS
KATHRIN BRINGMANN, JEREMY LOVEJOY, AND ROBERT OSBURN
Abstract.
We study two types of crank moments and two types of rank moments for overpartitions.We show that the crank moments and their derivatives, along with certain linear combinations of therank moments and their derivatives, can be written in terms of quasimodular forms. We then use thisfact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and7 for some combinatorial functions which may be expressed in terms of the second moments. Finally,we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz classnumber H ( n ). Introduction
Dyson’s rank of a partition is the largest part minus the number of parts [14]. Let N ( m, n ) denotethe number of partitions of n whose rank is m . The Andrews-Garvan crank is either the largest part, if1 does not occur, or the difference between the number of parts larger than the number of 1’s and thenumber of 1’s, if 1 does occur [1]. For n = 1 let M ( m, n ) denote the number of partitions of n whosecrank is m . Even though there is only one partition of one, for technical reasons we set M (0 ,
1) = − M ( − ,
1) = M (1 ,
1) = 1, and M ( m,
1) = 0 otherwise. Then the k th rank moment N k ( n ) and the k thcrank moment M k ( n ) are given by N k ( n ) := X m ∈ Z m k N ( m, n ) , (1.1)and M k ( n ) := X m ∈ Z m k M ( m, n ) . (1.2)Since their introduction by Atkin and Garvan [4], the rank and crank moments and their linearcombinations have been the subject of a number of works [2, 3, 5, 6, 16, 17]. A key role in severalof these studies is played by the fact that the crank moments and their derivatives, along with aspecific linear combination of the rank moments and their derivatives, can be expressed in terms ofquasimodular forms. Here we shall see that this holds in the case of overpartitions as well.Recall that an overpartition [13] is a partition in which the first occurrence of each distinct numbermay be overlined. For example, the 14 overpartitions of 4 are4 , , , , , , , , , , , , , . (1.3) Date : September 21, 2018.2000
Mathematics Subject Classification.
Primary: 11F11, 11P83; Secondary: 05A19.The first author was partially supported by NSF grant DMS-0757907 while the second and third authors were partiallysupported by a PHC Ulysses grant. Part of this paper was written while the first author was in residence at the Max-Planck Institute. She thanks the institute for providing a stimulating environment.
We denote by P the generating function for overpartitions (throughout q = e πiτ and τ = x + iy with x , y ∈ R ) [13], P = P ( q ) = Y n ≥ (1 + q n )(1 − q n ) . The case of overpartitions is somewhat different from that of partitions. First, there are two distinctranks of interest: Dyson’s rank and the M M ℓ ( · ) to denote the largest part of an object, n ( · ) to denote thenumber of parts, and λ o for the subpartition of an overpartition consisting of the odd non-overlinedparts. Then the M λ is M λ ) := (cid:24) ℓ ( λ )2 (cid:25) − n ( λ ) + n ( λ o ) − χ ( λ ) , where χ ( λ ) = 1 if the largest part of λ is odd and non-overlined and χ ( λ ) = 0 otherwise.Let N ( m, n ) (resp. N m, n )) denote the number of overpartitions of n whose rank (resp. M m . We define the rank moments N k ( n ) and N k ( n ), along with their generating functions R k and R k , by R k = R k ( q ) := X n ≥ N k ( n ) q n := X n ≥ X m ∈ Z m k N ( m, n ) ! q n , (1.4)and R k = R k ( q ) := X n ≥ N k ( n ) q n := X n ≥ X m ∈ Z m k N m, n ) ! q n . (1.5)We note that in light of the symmetries N ( m, n ) = N ( − m, n ) [19] and N m, n ) = N m, n ) [20], wehave R k = R k = 0 when k is odd.The second difference between partitions and overpartitions is that in the latter case no notion ofcrank has been defined. Indeed, the crank for partitions arose because of its relation to Ramanujan’scongruences, and Choi has shown that no such congruences exist for overpartitions [11]. What wewill be required to consider are two “residual cranks”. The first residual crank of an overpartition isobtained by taking the crank of the subpartition consisting of the non-overlined parts. The secondresidual crank is obtained by taking the crank of the subpartition consisting of all of the even non-overlined parts divided by two.Let M ( m, n ) (resp. M m, n )) denote the number of overpartitions of n with first (resp. second)residual crank equal to m . Here we make the appropriate modifications based on the fact that forpartitions we have M (0 ,
1) = − M ( − ,
1) = M (1 ,
1) = 1. For example, the overpartition7 + 5 + 2 + 1 contributes a − M (0 ,
15) and a +1 to M ( − ,
15) and M (1 , M k ( n ) and M k ( n ), along with their generating functions C k and C k , by C k = C k ( q ) := X n ≥ M k ( n ) q n := X n ≥ X m ∈ Z m k M ( m, n ) ! q n , (1.6)and C k = C k ( q ) := X n ≥ M k ( n ) q n := X n ≥ X m ∈ Z m k M m, n ) ! q n . (1.7)As with the rank moments, the crank moments turn out to be 0 for k odd (see (2.1) and (2.2)). ANK AND CRANK MOMENTS FOR OVERPARTITIONS 3
We are now ready to state the quasimodularity properties of the rank and crank moments foroverpartitions.
Theorem 1.1.
For k ≥ let W k denote the space of quasimodular forms on Γ (2) of weight at most k having no constant term. The following functions are in P · W k : ( i ) The functions in C k := (cid:8) δ mq (cid:0) C j (cid:1) : m ≥ , ≤ j ≤ k, j + m ≤ k (cid:9) , ( ii ) the functions in C k := (cid:8) δ mq (cid:0) C j (cid:1) : m ≥ , ≤ j ≤ k , j + m ≤ k (cid:9) , ( iii ) for a = 2 k the function (cid:0) a − a + 2 (cid:1) R a + 2 a/ − X i =1 (cid:18) a i (cid:19) (cid:0) i − i − (cid:1) δ q R a − i + a/ − X i =1 (cid:18)(cid:18) a i (cid:19) (2 i + 1) + 2 (cid:18) a i + 1 (cid:19) (cid:0) − i +1 (cid:1) + 12 (cid:18) a i + 2 (cid:19) (cid:0) i +2 − i +2 − (cid:1)(cid:19) R a − i , ( iv ) for a = 2 k the function (cid:0) a − a + 2 (cid:1) R a + 12 a/ − X i =1 (cid:18) a i (cid:19) (cid:0) i − i − (cid:1) δ q R a − i + a/ − X i =1 (cid:18)(cid:18) a i (cid:19) (2 i + 1) + 2 (cid:18) a i + 1 (cid:19) (cid:0) − i +1 (cid:1) + 12 (cid:18) a i + 2 (cid:19) (cid:0) i +2 − i +2 − (cid:1)(cid:19) R a − i . It turns out that for k = 2 ,
3, and 4 the number of functions above exceeds the dimension of W k , which implies relations among these functions. In Corollaries 3.1–3.3, we compute several suchrelations. This is the same approach taken by Atkin and Garvan in their study of rank and crankmoments of partitions [4].Then we show how Theorem 1.1 can be used to deduce congruence properties for combinatorialfunctions which can be expressed in terms of second rank and crank moments. First, let nov ( n ) (resp. ov ( n )) denote the sum, over all overpartitions of n , of the non-overlined (resp. overlined) parts. Forexample, (1.3) shows that ov (4) = 21 and nov (4) = 35. The generating functions of nov ( n ) and ov ( n )are given by (see Section 4) N ov ( q ) := ∞ X n =0 nov ( n ) q n = P ∞ X n =1 n q n − q n , (1.8) Ov ( q ) := ∞ X n =0 ov ( n ) q n = P ∞ X n =1 n q n q n . (1.9) Theorem 1.2.
We have ( n + 2) nov ( n ) ≡ ( n + 4 n + 3) ov ( n ) (mod 5) , (1.10) and ( n + 1) nov ( n ) ≡ (4 n − n − ov ( n ) (mod 7) . (1.11) KATHRIN BRINGMANN, JEREMY LOVEJOY, AND ROBERT OSBURN
Notice that congruences like (1.10) and (1.11) imply simpler congruences in arithmetic progressionsfor ov ( n ) and nov ( n ) modulo 5 and 7.Next, let spt n ) (resp. spt n )) denote the sum, over all overpartitions λ of n , of the numberof occurrences of the smallest part of λ , provided this smallest part is odd (resp. even). Let spt ( n )be the sum of these two functions. For example, using (1.3) we have spt spt spt (4) = 26. When the overpartition has no overlined parts, spt ( n ) reduces to Andrews’ smallest partsfunction spt ( n ) [3, 16, 17]. The functions spt n ) and spt ( n ) can be easily computed using (4.2) and(4.3). Theorem 1.3.
We have spt n ) ≡ spt n + 1) ≡ , (1.12) spt (3 n ) ≡ , (1.13) spt n + 3) ≡ , (1.14) and spt n ) ≡ . (1.15)To finish we use a different method to give a congruence modulo 3 between spt n ) and α ( n ), thenumber of overpartitions with even rank minus the number with odd rank. Theorem 1.4.
We have spt n ) ≡ (cid:18) n (cid:19) α ( n ) (mod 3) . Remark 1.5. In [7] , the coefficients α ( n ) are related to the Hurwitz class number H ( n ) of binaryquadratic forms of discriminant − n . To be more precise, it is shown in [7] that ( − n α ( n ) = − H (4 n ) if n ≡ , , − H ( n ) if n ≡ , − H ( n ) if n ≡ , − H ( n ) − r ( n/ if | n, (1.16) where r ( n ) is given by ∞ X n =0 r ( n ) q n := Θ ( τ ) , with Θ( τ ) := P n ∈ Z q n . It is well-known that r ( n ) = H (4 n ) if n ≡ , , H ( n ) if n ≡ ,r ( n/ if n ≡ , if n ≡ , (1.17) thus modulo , spt n ) is related to class numbers. As a corollary, class number relations imply the following multiplicative formula:
ANK AND CRANK MOMENTS FOR OVERPARTITIONS 5
Corollary 1.6.
Let ℓ = 2 , be a prime. Then we have spt (cid:0) ℓ n (cid:1) + (cid:18) − nℓ (cid:19) spt n ) + ℓ spt (cid:16) nℓ (cid:17) ≡ ( ℓ + 1) spt n ) (mod 3) . Remark 1.7.
Work of the authors [8] shows that the generating function for spt n ) can (up to aquasimodular form) be viewed as the holomorphic part of a harmonic Maass form (see Section 5 forthe definition). Corollary 1.6 now says that modulo the generating function for spt n ) is a Heckeeigenform. The paper is organized as follows. In Section 2, we recall some facts about quasimodular forms andprove Theorem 1.1. In Section 3, we compute some exact relations involving rank and crank moments.In Section 4, we write the combinatorial functions in Theorems 1.2 and 1.3 in terms of the rank andcrank moments and prove these theorems. In Section 5, we recall the notion of harmonic Maass formsalong with some results from [8], and prove Theorem 1.4 and Corollary 1.6.2.
Proof of Theorem 1.1
Before proving Theorem 1.1, we recall a few facts about quasimodular forms [18]. First, quasimod-ular forms on Γ ( N ) may be regarded as polynomials in the Eisenstein series E whose coefficients aremodular forms (of non-negative weight) on Γ ( N ). The reader unfamiliar with the theory of modularforms may consult [21]. Here we have E ( τ ) := 1 − X n ≥ nq n (1 − q n ) . Second, the space of quasimodular forms on Γ ( N ) is preserved by the differential operator δ q := q ddq .More specifically, this operator sends a quasimodular form of weight 2 k to a quasimodular form ofweight 2 k + 2. Finally, replacing q by q sends a quasimodular form of weight 2 k on Γ ( N ) to aquasimodular form of weight 2 k on Γ (2 N ). Proof of Theorem 1.1.
We now prove parts ( i ) and ( ii ) of Theorem 1.1. Let C ( z ; q ) denote the two-variable generating function for the crank of a partition, C ( z ; q ) := X m ∈ Z n ≥ M ( m, n ) z m q n = ( q ; q ) ∞ ( zq ; q ) ∞ ( q/z ; q ) ∞ . Here we employ the standard q -series notation,( a ; q ) ∞ := Y k ≥ (cid:16) − aq k (cid:17) . By definition, the residual cranks have two-variable generating functions C ( z ; q ) := X m ∈ Z n ≥ M ( m, n ) z m q n = ( − q ; q ) ∞ C ( z ; q ) = ( q ; q ) ∞ ( zq ; q ) ∞ ( q/z ; q ) ∞ , (2.1)and C z ; q ) := X m ∈ Z n ≥ M m, n ) z m q n = ( − q ; q ) ∞ ( q ; q ) ∞ C (cid:0) z ; q (cid:1) = ( − q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( zq ; q ) ∞ ( q /z ; q ) ∞ . (2.2) KATHRIN BRINGMANN, JEREMY LOVEJOY, AND ROBERT OSBURN
Now using the differential operator δ z := z ddz we have δ jz (cid:0) C ( z ; q ) (cid:1) (cid:12)(cid:12)(cid:12) z =1 = ( C j if j is even , j is odd , and δ jz (cid:0) C z ; q ) (cid:1) (cid:12)(cid:12)(cid:12) z =1 = ( C j if j is even , j is odd . But δ jz (cid:0) C ( z ; q ) (cid:1) = ( − q ; q ) ∞ δ jz ( C ( z ; q )) and Atkin and Garvan [4, Section 4] have already shown thatif j ≥
1, then δ jz ( C ( z ; q )) | z =1 is in the space P · W j , where P = P ( q ) := 1 / ( q ; q ) ∞ is the generatingfunction for partitions and W j is the space of quasimodular forms of weight at most 2 j on SL ( Z )having no constant term. Since P = ( − q ; q ) ∞ P , we have that C j is in P · W j . In a similar way wesee that C j is in P · W j .To finish we may calculate that δ q (cid:0) P (cid:1) = P X n ≥ nq n (1 − q n ) − X n ≥ nq n (1 − q n ) , and hence δ q ( P ) ∈ P · W . By the Leibniz rule and the fact that δ q maps the space W k into W k +1 ,we have that δ q f ∈ P · W k +1 if f ∈ P · W k . This completes the proof of parts ( i ) and ( ii ).For parts ( iii ) and ( iv ), we use partial differential equations established in [8]. Let R ( z ; q ) denotethe two-variable generating function for N ( m, n ), R ( z ; q ) := X m ∈ Z n ≥ N ( m, n ) z m q n . Thus we have δ jz (cid:0) R ( z ; q ) (cid:1) (cid:12)(cid:12)(cid:12) z =1 = ( R j if j is even , j is odd . We have the following partial differential equation which relates C ( z ; q ) and R ( z ; q ) [8]: z (1 + z ) ( q ) ∞ ( − q ) ∞ [ C ( z ; q )] ( − zq ; q ) ∞ ( − q/z ; q ) ∞ = (cid:16) − z ) (1 + z ) δ q + z (1 + z ) + 2 z (1 − z ) δ z + 12 (1 + z )(1 − z ) δ z (cid:17) R ( z ; q ) . (2.3)Let a be even and positive. After applying δ az to both sides of (2.3) and setting z = 1 we get1 P P a X j =0 (cid:18) aj (cid:19) δ jz (cid:8) ( z + z ) C ( z ; q ) (cid:9) δ a − jz { ( − zq ; q ) ∞ ( − q/z ; q ) ∞ }| z =1 − (2 a + 1) P − a − a − δ q ( P ) = (cid:0) a − a + 2 (cid:1) R a + 2 a/ − X i =1 (cid:18) a i (cid:19) (cid:0) i − i − (cid:1) δ q R a − i + a/ − X i =1 (cid:18)(cid:18) a i (cid:19) (2 i + 1) + 2 (cid:18) a i + 1 (cid:19) (cid:0) − i +1 (cid:1) + 12 (cid:18) a i + 2 (cid:19) (cid:0) i +2 − i +2 − (cid:1)(cid:19) R a − i . (2.4) ANK AND CRANK MOMENTS FOR OVERPARTITIONS 7
We claim that the left hand side of (2.4) is in P · W k , where 2 k = a . This is clearly true for thefinal term. For the first term, we have already noted that for j ≥ δ jz C ( z ; q ) | z =1 ∈ P · W j .As for ( − zq ; q ) ∞ ( − q/z ; q ) ∞ , we may compute that δ z (cid:16) ( − zq ; q ) ∞ ( − q/z ; q ) ∞ (cid:17) = z ∞ X m =1 q m zq m − z − ∞ X m =1 q m z − q m ! ( − zq ; q ) ∞ ( − q/z ; q ) ∞ = ∞ X m =1 ∞ X s =1 ( − s q ms ( z − s − z s ) ! ( − zq ; q ) ∞ ( − q/z ; q ) ∞ , and for j ≥ δ jz ∞ X m =1 ∞ X s =1 ( − s q ms (cid:0) z − s − z s (cid:1)!(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =1 = j is even , − ∞ X m =1 ∞ X s =1 ( − s s j q ms if j is odd . Then one can check that − ∞ X m =1 ∞ X s =1 ( − s s j q ms = − j +2 X n ≥ n j q n (1 − q n ) + 2 X n ≥ n j q n (1 − q n ) . Thus for j ≥ δ jz { ( − zq ; q ) ∞ ( − q/z ; q ) ∞ }| z =1 ∈ ( P /P ) · W j . Putting everything together we see that the only contribution from the first term on the left hand sidewhich is not in P · W k is 1 P P δ az (cid:8) ( z + z ) (cid:9) C ( z ; q ) ( − zq ; q ) ∞ ( − q/z ; q ) ∞ (cid:12)(cid:12)(cid:12)(cid:12) z =1 . But this cancels with the second term on the left hand side. This establishes part ( iii ).The proof of part ( iv ) is the same, except that we use the partial differential equation [8]2 z (1 + z ) (cid:0) q ; q (cid:1) ∞ (cid:2) C (cid:0) z ; q (cid:1)(cid:3) ( − zq ; q ) ∞ ( − q/z ; q ) ∞ = (cid:16) (1 + z )(1 − z ) δ q + 2 z (1 + z ) + 4 z (1 − z ) δ z + (1 + z )(1 − z ) δ z (cid:17) R z ; q ) . Here R z ; q ) is the two-variable generating function for N m, n ), so that δ jz (cid:0) R z ; q ) (cid:1) (cid:12)(cid:12)(cid:12) z =1 = ( R j if j is even , j is odd . (cid:3) KATHRIN BRINGMANN, JEREMY LOVEJOY, AND ROBERT OSBURN Exact relations
From part ( b ) of Proposition 1 in [18] and known formulas for the dimensions of spaces of modularforms on Γ (2) (see [21]), we have that the sequence { dim ( W k ) } ∞ k =1 begins { , , , , , , . . . } .Suppose first that k = 2. Then there are 6 functions in parts ( i ) and ( ii ) of Theorem 1.1. Computation(with MAPLE, for example) shows that they are linearly independent. Hence, each function in parts( iii ) and ( iv ) may be written as a linear combination of these 6 functions, and we compute thefollowing: Corollary 3.1.
We have N ( n ) = ( − n − N ( n ) + (cid:18) −
216 + 24 n (cid:19) M ( n ) + 19277 M ( n ) + (cid:18)
260 + 184 n (cid:19) M ( n ) − M ( n )(3.1) and N ( n ) = ( − n − N ( n ) + (cid:18) −
27 + 3 n (cid:19) M ( n ) + 2477 M ( n ) + (cid:18) − n (cid:19) M ( n ) − M ( n ) . (3.2)Now let k = 3. Again we find that the 12 functions in parts ( i ) and ( ii ) of Theorem 1.1 are linearlyindependent, and so the functions in parts ( iii ) and ( iv ) may be written in terms of them. Followingthe lead of Atkin and Garvan, we use (3.1) and (3.2) to eliminate N ( n ) and N ( n ), thus expressing N ( n ) (resp. N ( n )) in terms of N ( n ) (resp. N ( n )) and the crank moments. Corollary 3.2.
We have N ( n ) = (cid:0) n + 48 n (cid:1) N ( n ) + (cid:18) n − n (cid:19) M ( n )+ (cid:18) − − n (cid:19) M ( n ) + (cid:18) (cid:19) M ( n )+ (cid:18) − − n − n (cid:19) M ( n )+ (cid:18) n (cid:19) M ( n ) + (cid:18) − (cid:19) M ( n ) (3.3) and N ( n ) = (cid:0) n + 3 n (cid:1) N ( n ) + (cid:18) n − n (cid:19) M ( n )+ (cid:18) − − n (cid:19) M ( n ) + (cid:18) (cid:19) M ( n )+ (cid:18) − n n (cid:19) M ( n )+ (cid:18) n (cid:19) M ( n ) + (cid:18) − (cid:19) M ( n ) . (3.4)Now for k = 4, there are 22 functions in Theorem 1.1 and the dimension of P · W k is 21. Thisimplies a relation among these 22 functions. If we would like relations wherein only one type of rank ANK AND CRANK MOMENTS FOR OVERPARTITIONS 9 moment occurs then we may combine the function F = F ( q ) := q ( q ; q ) ∞ (cid:0) q ; q (cid:1) ∞ := X n ≥ a F ( n ) q n with the functions in C and C to get a basis for P · W . (The fact that F is in this space followsfrom the fact that q ( q ; q ) ∞ (cid:0) q ; q (cid:1) ∞ is a cusp form of weight 8 on Γ (2)). Then each of the functionsin ( iii ) and ( iv ) of Theorem 1.1 may expressed in terms of this basis. We display the relation for thecase of N k ( n ), again using results above to eliminate the 4th and 6th rank moments in favor of the2nd. Corollary 3.3. N ( n ) = (cid:0) − − n − n − n (cid:1) N ( n ) + (cid:18) (cid:19) a F ( n )+ (cid:18) − − n − n n (cid:19) M ( n )+ (cid:18) n n (cid:19) M ( n )+ (cid:18) − − n (cid:19) M ( n ) + (cid:18) (cid:19) M ( n )+ (cid:18) n n n (cid:19) M ( n )+ (cid:18) − − n − n (cid:19) M ( n )+ (cid:18) n (cid:19) M ( n ) + (cid:18) − (cid:19) M ( n ) . (3.5)When k ≥
5, the number of functions in Theorem 1.1 is smaller than the dimension of P · W k .Presumably this could be remedied by adding functions of the form P f , where f is a cusp form, alongwith their δ q - derivatives. We shall not pursue this here.4. Proof of Theorems 1.2 and 1.3
We begin this section by expressing our combinatorial functions in terms of the second moments N ( n ), N ( n ), M ( n ), and M ( n ). Proposition 4.1.
We have nov ( n ) = M ( n ) and ov ( n ) = M ( n ) − M ( n ) .Proof. Dyson [15] has shown that M ( n ) = 2 np ( n ), where p ( n ) is the number of partitions of n . Since X n ≥ M ( n ) q n = δ z C ( z ; q ) (cid:12)(cid:12) z =1 , we have that X n ≥ M ( n ) q n = ( − q ; q ) ∞ δ z C ( z ; q ) | z =1 = ( − q ; q ) ∞ X n ≥ np ( n ) q n = 2 X n ≥ nov ( n ) q n . Similarly, we find that M ( n ) may be interpreted as enov ( n ), where enov ( n ) denotes the sum, overall overpartitions of n , of the even non-overlined parts. Using Euler’s identity between the numberof partitions of n into odd parts and the number of partitions of n into distinct parts, we see that nov ( n ) − enov ( n ) = ov ( n ). (cid:3) Note that by applying δ q to P , we see that1( q ; q ) ∞ X n ≥ nq n (1 − q n ) = X n ≥ np ( n ) q n , (4.1)which gives equations (1.8) and (1.9).We now turn to the smallest parts functions. Proposition 4.2.
We have spt ( n ) = M ( n ) − N ( n ) and spt n ) = M ( n ) − N ( n ) .Proof. By the work in [8], we find that ∞ X n =0 spt ( n ) q n = ( − q ; q ) ∞ ( q ; q ) ∞ X n ≥ nq n (1 − q n ) − ∞ X n =0 N ( n ) q n , (4.2)and ∞ X n =0 spt n ) q n = ( − q ; q ) ∞ ( q ; q ) ∞ X n ≥ nq n (1 − q n ) − ∞ X n =0 N ( n ) q n . (4.3)Combining (4.1) with (4.2), (4.3), and the proof of Proposition 4.1 finishes the proof. (cid:3) We now prove the congruences in Theorems 1.2 and 1.3.
Proof of Theorem 1.2.
For (1.10), we simply multiply (3.3) by 5 and reduce modulo 5. The result is (cid:0) n + n + 2 (cid:1) M ( n ) + (cid:0) n + 4 n + 3 (cid:1) M ( n ) ≡ , (4.4)which implies the desired congruence.For (1.11), we first multiply (3.1) by 7 and reduce modulo 7. The result is(2 + 6 n ) M ( n ) + 6 M ( n ) + (2 + 4 n ) M ( n ) ≡ . (4.5)Next we take the set C ∪ C ∪ { F } and replace δ q C by C C /P and δ q C by C C /P . This turnsout to be a basis for P · W . Expressing the function in part ( iii ) of Theorem 1.1 in terms of thisbasis, multiplying by 7 and reducing modulo 7 gives (cid:0) n + 2 n + 3 n (cid:1) M ( n ) + 6 M ( n ) + (cid:0) n + 5 n + n (cid:1) M ( n ) ≡ . Using (4.5) to substitute for M ( n ) gives (cid:0) n + 3 n + 3 (cid:1) M ( n ) ≡ (cid:0) n + 3 n + 3 (cid:1) M ( n ) (mod 7) , ANK AND CRANK MOMENTS FOR OVERPARTITIONS 11 and the congruence (1.11) follows. (cid:3)
Proof of Theorem 1.3.
First reduce (3.2) modulo 3 to obtain(2 n + 2) N ( n ) ≡ (2 n + 2) M ( n ) (mod 3) . Since spt n ) = M ( n ) − N ( n ), we have (1.12).Reducing (3.1) modulo 3 we obtain(2 n + 2) N ( n ) ≡ (2 n + 2) M ( n ) (mod 3) . Combined with the fact that nov (3 n ) ≡ − ov (3 n ) (mod 3) (since nov ( n ) + ov ( n ) = np ( n )) and thefact that spt ( n ) = M ( n ) − N ( n ), we have (1.13).Next we perform the same computation used to obtain (3.4), except that we replace δ q C by C C /P . Reducing the result modulo 5 gives (cid:0) − n (cid:1) N ( n ) ≡ (cid:0) n + 3 (cid:1) M ( n ) (mod 5) . (4.6)Combining this with (4.4) when n is replaced by 5 n + 3 gives (1.14).Finally we perform the same calculation used to obtain (3.3), again replacing δ q C by C C /P .Reducing the result modulo 5 gives (cid:0) n (cid:1) N ( n ) ≡ (cid:0) n + 4 n (cid:1) M ( n ) + (4 + 4 n ) M ( n ) (mod 5) . Combining this with (4.6) and (4.4) when n is replaced by 5 n , together with the fact that spt n ) = M ( n ) − N ( n ) − M ( n ) + N ( n ), gives (1.15). (cid:3) Proof of Theorem 1.4 and Corollary 1.6
Proof of Theorem 1.4.
Let
Spt
Spt q ) denote the generating function for spt n ) and let f = f ( q )denote the generating function for α ( n ). Since by (1.12) and (1.13) we have spt n ) ≡ , to prove Theorem 1.4 it is enough to show that G = G ( q ) := (cid:18) • (cid:19) ⊗ (cid:18) Spt − (cid:18) • (cid:19) ⊗ f (cid:19) ≡ , where for a character χ and a q -series g , χ ⊗ g denotes the twist of g by χ , i.e., the n th Fouriercoefficient of g is multiplied by χ ( n ). Let us next recall the definition of a harmonic Maass form. If k ∈ Z \ Z , then the weight k hyperbolic Laplacian is given by∆ k := − y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + iky (cid:18) ∂∂x + i ∂∂y (cid:19) . (5.1)If v is odd, then define ǫ v by ǫ v := ( v ≡ ,i if v ≡ . (5.2)Moreover we let χ be a Dirichlet character. A harmonic Maass form of weight k with Nebentypus χ on a subgroup Γ ⊂ Γ (4) is any smooth function g : H → C satisfying the following: (1) For all A = (cid:0) a bc d (cid:1) ∈ Γ and all τ ∈ H , we have g ( Aτ ) = (cid:18) cd (cid:19) k ǫ − kd χ ( d ) ( cτ + d ) k g ( τ ) . (2) We have that ∆ k g = 0.(3) The function g ( τ ) has at most linear exponential growth at all the cusps of Γ.Define the integral N H ( τ ) := 12 √ πi Z i ∞− ¯ τ η ( u ) η (2 u )( − i ( τ + u )) du, where η ( τ ) is Dedekind’s eta function. Combining (4.2) and (4.3) with Theorems 4.1 and 5.1 of [8],we may conclude that M ( τ ) := Spt η (2 τ ) η ( τ ) E ( τ ) − η (2 τ ) η ( τ ) E (2 τ ) + N H ( τ )is a weight harmonic Maass form on Γ (16). From [7] we have that M ( τ ) := f − N H ( τ )is also a harmonic Maass form of weight on Γ (16).Turning back to the proof of Theorem 1.4, it is not hard to check that G ≡ H (mod 3) , where H = H ( q ) := (cid:18) • (cid:19) ⊗ (cid:18) (cid:18) Spt η (2 τ ) η ( τ ) E ( τ ) − η (2 τ ) η ( τ ) E (2 τ ) (cid:19) + η (2 τ ) η ( τ ) + η (2 τ )3 η ( τ ) ( − E (2 τ ) + E ( τ )) − (cid:18) • (cid:19) ⊗ f (cid:19) . As in the proof of Proposition 4.1 of [9], one can show that the non-holomorphic parts of M ( τ ) and M ( τ ) are supported on negative squares. This easily yields that H is a linear combination of weaklyholomorphic modular forms, i.e. meromorphic modular forms whose poles are supported in the cusps,of weights − , , and on Γ (144). We next place bounds on the orders of vanishing of H in thecusps. Clearly E ( τ ) and E (2 τ ) have no poles. Moreover from the transformation law of f (see [7])it follows that f also has no poles. Using this one can show that poles can only arise from η (2 τ ) η ( τ ) andthus are of the form ac with c odd. Using properties of twists, we can bound the orders of vanishingof H at ac with c odd as follows: If 9 | c , its order can be bounded by − , if 3 k c its order is boundedby − , and if 3 ∤ c the order is bounded by − . This now easily yields that η ( τ ) η (2 τ ) G is the sum ofthree holomorphic modular forms of weight 4, 6, and 8, respectively. Using the fact that η ( τ ) η (3 τ ) is aholomorphic weight 2 modular form on Γ (9) satisfying η ( τ ) η (3 τ ) ≡ , ANK AND CRANK MOMENTS FOR OVERPARTITIONS 13 we can show that G is congruent to a holomorphic modular form of weight 8 on Γ (144) modulo 3.Sturm’s Theorem now gives that this form is congruent to 0 if the first (cid:20)
812 [SL ( Z ) : Γ (144)] (cid:21) + 1 = 193coefficients are congruent 0 modulo 3. This can be done by MAPLE. (cid:3) Corollary 1.6 now follows easily from Theorem 1.4 and the following
Proposition 5.1.
Let ℓ = 2 , . Then we have α (cid:0) ℓ n (cid:1) + (cid:18) − nℓ (cid:19) α ( n ) + ℓ α (cid:16) nℓ (cid:17) = ( ℓ + 1) α ( n ) . (5.3) Proof.
To prove (5.3), we have to show that g ℓ ( τ ) := f | T ℓ − ( ℓ + 1) f = 0 , where T ℓ is the usual half-integral weight Hecke-operator. Using that η ( τ ) η (2 τ ) is a Hecke eigenform witheigenvalue 1 + ℓ , one obtains from [10] that g ℓ ( τ ) is a weakly holomorphic modular form of weight on Γ (16). Since the coefficients of ¯ f have only polynomial growth it is a holomorphic form. Thevalence formula now gives that g ℓ = 0 if its first 4 Fourier coefficients equal 0. Thus to finish theproof, we have to show that (5.3) is true for 0 ≤ n ≤
3. For n = 0 this claim is trivial. For the othercases recall (1.16) and (1.17). Moreover we need the fact [12] that if − n = Df , where D is a negativefundamental discriminant, then H ( n ) = h ( D ) w ( D ) X d | f µ ( d ) (cid:18) Dd (cid:19) σ ( f /d ) . (5.4)Here h ( D ) is the class number of Q ( √ D ), w ( D ) is half the number of units in the ring of integers of Q ( √ D ), σ ( n ) is the sum of the divisors of n , and µ ( n ) is the M¨obius function. We only show (5.3)for n = 1 since the other cases follow similarly. In this case we have to show that α (cid:0) ℓ (cid:1) = 2 (cid:18) ℓ + 1 − (cid:18) − ℓ (cid:19)(cid:19) . Firstly we have from (1.16) that α (cid:0) ℓ (cid:1) = 4 H (cid:0) ℓ (cid:1) . Since h ( −
4) = 1 and ω ( −
4) = 2, (5.4) yields α (cid:0) ℓ (cid:1) = 2 (cid:18) σ ( ℓ ) − (cid:18) − ℓ (cid:19)(cid:19) = 2 (cid:18) ℓ + 1 − (cid:18) − ℓ (cid:19)(cid:19) , as claimed. (cid:3) References [1] G.E. Andrews and F. Garvan, Dyson’s crank of a partition,
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School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U. S. A.CNRS, LIAFA, Universit´e Denis Diderot, 2, Place Jussieu, Case 7014, F-75251 Paris Cedex 05, FRANCESchool of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
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