Weighted partition rank and crank moments. I. Andrews{Beck type congruences Shane Chern Abstract. Recently, George Beck conjectured and George Andrews proved a number of interesting congruences concerning Dyson's rank function and Andrews{Garvan's crank function. In this paper, we shall prove two identities involving the weighted rank and crank moments, from which we may deduce more congruences of Andrews{Beck type. Keywords. Partitions, Dyson's rank, Andrews{Garvan's crank, Andrews{Beck type con- gruences. 2010MSC. 11P83, 11P81, 05A19. 1. Introduction 1.1. Background. As usual, a partition of a positive integer n is a weakly de- creasing sequence of positive integers whose sum equals n. For example, 4 has five partitions: 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. If we denote by p(n) the number of partitions of n, then p(4) = 5. In the theory of partitions, one of the most well-known results is achieved by Ramanujan [1, Chapter 10]. He found that p(n) satisfies the following three con- gruences: p(5n + 4) ≡ 0 (mod 5); (1.1) p(7n + 5) ≡ 0 (mod 7) (1.2) and p(11n + 6) ≡ 0 (mod 11): (1.3) In his famous 1944 paper [7], Dyson defined the rank of a partition to be the largest part minus the number of parts. Dyson then conjectured that the rank function can provide a combinatorial interpretation of (1.1) and (1.2); this was later confirmed by Atkin and Swinnerton-Dyer [6]. Dyson also conjectured the existence of a crank function, which can combinatorially interpret all the three Ramanujan congruences. Such a function was not found until over four decades later by Andrews and Garvan [4]. Following [6], we define by N(m; k; n) the number of partitions of n with rank congruent to m modulo k. Atkin and Swinnerton-Dyer proved (1.1) and (1.2) by showing that for 0 ≤ i ≤ 4, 1 N(i; 5; 5n + 4) = p(5n + 4) 5 2 S. Chern and that for 0 ≤ i ≤ 6, 1 N(i; 7; 7n + 5) = p(7n + 5): 7 In a recent paper [3], Andrews mentioned that George Beck has made a number of new conjectures along this line. Instead of considering the N(m; k; n) function, Beck studied the total number of parts in the partitions of n with rank congruent to m modulo k, which is defined by NT (m; k; n). One of the results proved by Andrews reads as follows. Theorem. If i = 1 or 4, then for all n ≥ 0, NT (1; 5; 5n + i) + 2NT (2; 5; 5n + i) − 2NT (3; 5; 5n + i) − NT (4; 5; 5n + i) ≡ 0 (mod 5): (1.4) Andrews further remarked that (1.4) is trivial if one replaces the NT (m; k; n) function by the rank function N(m; k; n). This is simply due to the symmetry N(m; k; n) = N(k − m; k; n): However, the above phenomenon is generally false for the NT (m; k; n) function. Hence, the validness of (1.4) is to some extent exciting. In his proof of (1.4), Andrews did not apply the differentiation technique directly to the three-variable generating function X X x](λ)zrank(n)qn n≥0 λ`n where the inner summation runs through all partitions of n and ](λ) and rank(λ) respectively represent the number of parts in λ and the rank of λ. Instead, he transformed the generating function to a tractable representation so that he can use some results proved already in [6]. However, we observe that if we study a weighted rank moment, then we may obtain a connection with the second Atkin{Garvan rank moment [5] so that (1.4) follows as an immediate consequence. Likewise, a weighted crank moment will imply some new conjectures of Beck concerning the crank function. We will state our results in x1.3. 1.2. Notation and terminology. Let P be the set of partitions. Again, the notation λ ` n means that λ is a partition of n. Below let λ always be a partition. Let jλj be the size of λ. We use ](λ) and !(λ) to denote the number of parts in λ and the number of ones in λ, respectively. Further, rank(λ) and crank(λ) denotes the rank and crank of λ. As defined previously, NT (m; k; n) denotes the total number of parts in the partitions of n with rank congruent to m modulo k. We also denote by N(m; n) the number of partitions of n whose rank is m. Then the second Atkin{Garvan rank moment N2(n) is defined by X1 2 N2(n) := m N(m; n): m=−∞ Further, M!(m; k; n) counts the total number of ones in the partitions of n with crank congruent to m modulo k. Weighted partition rank and crank moments 3 Finally, we adopt the standard q-series notation nY−1 m (A; q)n := (1 − Aq ) m=0 and Y1 m (A; q)1 := (1 − Aq ): m=0 1.3. Main results. Our first result deals with the following weighted rank mo- ment. Theorem 1.1. We have 2 X X qn Xn qm ](λ) rank(λ)qjλj = − : (1.5) (q; q)2 (1 − qm)2 λ2P n≥1 n m=1 It follows that X 1 ](λ) rank(λ) = − N (n): (1.6) 2 2 λ`n Remark. It is worth pointing out that the following generating function identity for N2(n) is used most frequently. X X − n n(3n+1)=2 n n 2 ( 1) q (1 + q ) N2(n)q = − : (q; q)1 (1 − qn)2 n≥0 n≥1 See Eq. (3.4) in [2]. Likewise, for the weighted crank moment, our result reads as follows. Theorem 1.2. We have X 1 X qn !(λ) crank(λ)qjλj = − : (1.7) (q; q)1 (1 − qn)2 λ2P n≥1 It follows that X !(λ) crank(λ) = −np(n): (1.8) λ`n Remark. Let M(m; n) denote number of partitions of n with crank m. Atkin and Garvan [5] defined the k-th crank moment Mk(n) by X1 k Mk(n) = m M(m; n): m=−∞ It can be shown by means of a relation due to Dyson [8] that M2(n) = 2np(n): It turns out that X 1 !(λ) crank(λ) = − M (n): (1.9) 2 2 λ`n This is analogous to (1.6). 4 S. Chern As consequences of Theorems 1.1 and 1.2, we have a number of Andrews{Beck type congruences, including (1.4). Corollary 1.3. If i = 1 or 4, then for all n ≥ 0, NT (1; 5; 5n + i) + 2NT (2; 5; 5n + i) − 2NT (3; 5; 5n + i) − NT (4; 5; 5n + i) ≡ 0 (mod 5): (1.10) Corollary 1.4. If i = 1 or 5, then for all n ≥ 0, NT (1; 7; 7n + i) + 2NT (2; 7; 7n + i) + 3NT (3; 7; 7n + i) − 3NT (4; 7; 7n + i) − 2NT (5; 7; 7n + i) − NT (6; 7; 7n + i) ≡ 0 (mod 7): (1.11) Corollary 1.5. For all n ≥ 0, M!(1; 5; 5n + 4) + 2M!(2; 5; 5n + 4) − 2M!(3; 5; 5n + 4) − M!(4; 5; 5n + 4) ≡ 0 (mod 5): (1.12) Corollary 1.6. For all n ≥ 0, M!(1; 7; 7n + 5) + 2M!(2; 7; 7n + 5) + 3M!(3; 7; 7n + 5) − 3M!(4; 7; 7n + 5) − 2M!(5; 7; 7n + 5) − M!(6; 7; 7n + 5) ≡ 0 (mod 7): (1.13) Corollary 1.7. For all n ≥ 0, M!(1; 11; 11n + 6) + 2M!(2; 11; 11n + 6) + 3M!(3; 11; 11n + 6) + 4M!(4; 11; 11n + 6) + 5M!(5; 11; 11n + 6) − 5M!(6; 11; 11n + 6) − 4M!(7; 11; 11n + 6) − 3M!(8; 11; 11n + 6) − 2M!(9; 11; 11n + 6) − M!(10; 11; 11n + 6) ≡ 0 (mod 11): (1.14) At last, let spt(n) denote the total number of appearances of the smallest parts in all of the partitions of n. In [2], it was shown that 1 spt(n) = np(n) − N (n): 2 2 In view of (1.6) and (1.8), we immediately obtain the following interesting relation. Corollary 1.8. For all n ≥ 0, X X spt(n) = ](λ) rank(λ) − !(λ) crank(λ): (1.15) λ`n λ`n 2. The weighted rank moment We first study the weighted rank moment. It was shown in [3] that 2 X X X xnqn x](λ)zrank(λ)qn = : (2.1) (zq; q)n(xq=z; q)n n≥0 λ`n n≥0 Weighted partition rank and crank moments 5 Proof of Theorem 1.1. We first apply the operator [@=@x] to (2.1). X X x=1 ](λ)zrank(λ)qn ≥ ` n 0 λ n " # 2 X @ xnqn = ≥ @x (zq; q)n(xq=z; q)n n 0 " x=1 # 2 ( ) X xnqn @ xn = log ≥ (zq; q)n(xq=z; q)n @x (xq=z; q)n n 0 " x=1 !# 2 X qn @ Xn = n log x − log(1 − xqm=z) ≥ (zq; q)n(q=z; q)n @x n 0 " m=1# x=1 2 X qn n Xn qm = + m (zq; q)n(q=z; q)n x z − xq n≥0 m=1 x=1 ! 2 X qn Xn qm = n + m : (2.2) (zq; q)n(q=z; q)n z − q n≥1 m=1 We next make the following easy observation: for any n ≥ 1 where n can also be 1, [ ( )] " ( )# @ 1 Xn qm qm log = m + m 2 = 0: (2.3) @z (zq; q)n(q=z; q)n 1 − zq zq − z z=1 m=1 z=1 Applying the operator [@=@z] to (2.2) and using (2.3) yields X X z=1 ](λ) rank(λ)qn ≥ ` n 0 λ n " !# 2 X @ qn Xn qm = n + − m ≥ @z (zq; q)n(q=z; q)n z q n 1 " m=1 z=1# 2 ( ) X nqn @ 1 = log ≥ (zq; q)n(q=z; q)n @z (zq; q)n(q=z; q)n n 1 " z=1 # 2 ( ) X Xn qn qm @ 1 + m log m (zq; q)n(q=z; q)n z − q @z (zq; q)n(q=z; q)n(z − q ) n≥1 m=1 z=1 2 X qn Xn qm = − : (2.4) (q; q)2 (1 − qm)2 n≥1 n m=1 This is the first part of Theorem 1.1.