Weighted Partition Rank and Crank Moments. I. Andrews–Beck Type Congruences

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 congruences. 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 congruences:

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♯(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 §1.3. 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 |λ| be the size of λ. We use ♯(λ) and ω(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 ∑∞ 2 N2(n) := m N(m, n). m=−∞

Further, Mω(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 n∏−1 m (A; q)n := (1 − Aq ) m=0 and ∏∞ m (A; q)∞ := (1 − Aq ). m=0 1.3. Main results. Our ﬁrst result deals with the following weighted rank moment. Theorem 1.1. We have 2 ∑ ∑ qn ∑n qm ♯(λ) rank(λ)q|λ| = − . (1.5) (q; q)2 (1 − qm)2 λ∈P n≥1 n m=1 It follows that ∑ 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. ∑ ∑ − n n(3n+1)/2 n n 2 ( 1) q (1 + q ) N2(n)q = − . (q; q)∞ (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 ∑ 1 ∑ qn ω(λ) crank(λ)q|λ| = − . (1.7) (q; q)∞ (1 − qn)2 λ∈P n≥1 It follows that ∑ ω(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 ∑∞ 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 ∑ 1 ω(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,

Mω(1, 5, 5n + 4) + 2Mω(2, 5, 5n + 4)

− 2Mω(3, 5, 5n + 4) − Mω(4, 5, 5n + 4) ≡ 0 (mod 5). (1.12) Corollary 1.6. For all n ≥ 0,

Mω(1, 7, 7n + 5) + 2Mω(2, 7, 7n + 5)

+ 3Mω(3, 7, 7n + 5) − 3Mω(4, 7, 7n + 5)

− 2Mω(5, 7, 7n + 5) − Mω(6, 7, 7n + 5) ≡ 0 (mod 7). (1.13) Corollary 1.7. For all n ≥ 0,

Mω(1, 11, 11n + 6) + 2Mω(2, 11, 11n + 6) + 3Mω(3, 11, 11n + 6)

+ 4Mω(4, 11, 11n + 6) + 5Mω(5, 11, 11n + 6) − 5Mω(6, 11, 11n + 6)

− 4Mω(7, 11, 11n + 6) − 3Mω(8, 11, 11n + 6) − 2Mω(9, 11, 11n + 6)

− Mω(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, ∑ ∑ spt(n) = ♯(λ) rank(λ) − ω(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 ∑ ∑ ∑ 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=1 ♯(λ)zrank(λ)qn ≥ ⊢ n 0 λ n [ ] 2 ∑ ∂ xnqn = ≥ ∂x (zq; q)n(xq/z; q)n n 0 [ x=1 ] 2 ( ) ∑ xnqn ∂ xn = log ≥ (zq; q)n(xq/z; q)n ∂x (xq/z; q)n n 0 [ ( x=1 )] 2 ∑ qn ∂ ∑n = n log x − log(1 − xqm/z) ≥ (zq; q)n(q/z; q)n ∂x n 0 [ m=1] x=1 2 ∑ qn n ∑n qm = + − m ≥ (zq; q)n(q/z; q)n x z xq n 0 ( m=1 ) x=1 2 ∑ qn ∑n 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 ∑n 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 ∑ ∑ z=1 ♯(λ) rank(λ)qn ≥ ⊢ n 0 λ n [ ( )] 2 ∑ ∂ qn ∑n qm = n + − m ≥ ∂z (zq; q)n(q/z; q)n z q n 1 [ m=1 z=1] 2 ( ) ∑ nqn ∂ 1 = log ≥ (zq; q)n(q/z; q)n ∂z (zq; q)n(q/z; q)n n 1 [ z=1 ] 2 ( ) ∑ ∑n 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 ∑ qn ∑n qm = − . (2.4) (q; q)2 (1 − qm)2 n≥1 n m=1 This is the first part of Theorem 1.1. [ ( )] ∂ ∂ If one applies the operator ∂z z ∂z z=1 to the generating function 2 ∑ ∑ ∑ qn zrank(λ)qn = , (zq; q)n(q/z; q)n n≥0 λ⊢n n≥0 then one shall find that ∑ ∑ ∑ n 2 n N2(n)q = rank(λ) q n≥0 n≥0 λ⊢n 6 S. Chern [ ( )] 2 ∑ ∂ ∂ qn = z ≥ ∂z ∂z (zq; q)n(q/z; q)n n 0 [ z=1 ] 2 ( ) ∑ ∂ zqn ∑n qm qm = m + m 2 ∂z (zq; q)n(q/z; q)n 1 − zq zq − z n≥0 m=1 z=1 2 ∑ qn ∑n qm = 2 . (2.5) (q; q)2 (1 − qm)2 n≥1 n m=1 This combining with (2.4) gives the second part of Theorem 1.1. □

The arithmetic properties satisﬁed by N2(n) are extensively studied. For example, it was indicated on p. 285 of [9] that

N2(5n + 1) ≡ N2(5n + 4) ≡ 0 (mod 5) and

N2(7n + 1) ≡ N2(7n + 5) ≡ 0 (mod 7). On the other hand, it is trivial to see that ∑ ∑ ( ♯(λ) rank(λ)q|λ| ≡ NT (1, 5, n) + 2NT (2, 5, n) ∈P ≥ λ n 0 ) − 2NT (3, 5, n) − NT (4, 5, n) qn (mod 5) and ∑ ∑ ( ♯(λ) rank(λ)q|λ| ≡ NT (1, 7, n) + 2NT (2, 7, n) λ∈P n≥0 + 3NT (3, 7, n) − 3NT (4, 7, n) ) − 2NT (5, 7, n) − NT (6, 7, n) qn (mod 7). Corollaries 1.3 and 1.4 follow immediately.

3. The weighted crank moment Recall that the crank of a partition λ is deﬁned by { ℓ(λ) if ω(λ) = 0 crank(λ) = µ(λ) − ω(λ) if ω(λ) > 0, where ω(λ) denotes the number of ones in λ as before, ℓ(λ) denotes the largest part in λ and µ(λ) denotes the number of parts in λ larger than ω(λ). Notice that the crank for 1 is a little bit diﬀerent; one may consult [4] for details. Following [4], we have ∑ ∑ − ∑ j j −j ω(λ) crank(λ) n 1 q x q z x z q = + 2 j+1 (zq; q)∞ (q ; q)j−1(zq ; q)∞ n≥0 λ⊢n j≥1 ( ) 1 − q ∑ (zq; q) xq j = j (zq; q)∞ (q; q)j z j≥0 Weighted partition rank and crank moments 7

2 (1 − q)(xq ; q)∞ = . (3.1) (zq; q)∞(xq/z; q)∞ Here in the last identity we use the q-binomial theorem (see Theorem 2.1 in [1]): ∑ n (a; q) t (at; q)∞ n = . (q; q)n (t; q)∞ n≥0 If we take x = 1 in (3.1), then we recover the two-variable generating function in [4].

Proof of Theorem 1.2. This time we apply the operator [∂/∂x]x=1 to (3.1). ∑ ∑ ω(λ)zcrank(λ)qn ≥ ⊢ n 0 λ[n ] 2 ∂ (1 − q)(xq ; q)∞ = ∂x (zq; q)∞(xq/z; q)∞ [ x=1( )] 2 2 (1 − q)(xq ; q)∞ ∂ (xq ; q)∞ = log (zq; q)∞(xq/z; q)∞ ∂x (xq/z; q)∞  x=1  ∑ (q; q)∞ ∂ ( ) =  log(1 − xqn+1) − log(1 − xqn/z)  (zq; q)∞(q/z; q)∞ ∂x n≥1 ( ) x=1 ∑ n+1 n (q; q)∞ q q /z = − + . (3.2) (zq; q)∞(q/z; q)∞ 1 − qn+1 1 − qn/z n≥1

We then apply the operator [∂/∂z]z=1 to (3.2) and use (2.3) to deduce ∑ ∑ ω(λ) crank(λ)qn ≥ ⊢ n 0 λn  ( ) ∑ n+1 n ∂ (q; q)∞ q q /z =  − +  ∂z (zq; q)∞(q/z; q)∞ 1 − qn+1 1 − qn/z n≥1 [ ] z=1 ∑ n (q; q)∞ q /z 1 = · · (zq; q)∞(q/z; q)∞ 1 − qn/z qn − z n≥1 z=1 1 ∑ qn = − . (3.3) (q; q)∞ (1 − qn)2 n≥1

To prove the second part of Theorem 1.2, we simply observe that [ ] ∑ ∂ 1 np(n)qn = ∂z (zq; zq)∞ n≥0 z=1 1 ∑ nqn = (q; q)∞ 1 − qn n≥1 1 ∑ qn = . (q; q)∞ (1 − qn)2 n≥1 In view of (3.3), we arrive at the desired result. □ 8 S. Chern

Once again, we notice that ∑ ∑ ( |λ| ω(λ) crank(λ)q ≡ Mω(1, 5, n) + 2Mω(2, 5, n) ∈P ≥ λ n 0 ) n − 2Mω(3, 5, n) − Mω(4, 5, n) q (mod 5), ∑ ∑ ( |λ| ω(λ) crank(λ)q ≡ Mω(1, 7, n) + 2Mω(2, 7, n) λ∈P n≥0 + 3M (3, 7, n) − 3M (4, 7, n) ω ω ) n − 2Mω(5, 7, n) − Mω(6, 7, n) q (mod 7) and ∑ ∑ ( |λ| ω(λ) crank(λ)q ≡ Mω(1, 11, n) + 2Mω(2, 11, n) + 3Mω(3, 11, n) λ∈P n≥0

+ 4Mω(4, 11, n) + 5Mω(5, 11, n) − 5Mω(6, 11, n) − 4M (7, 11, n) − 3M (8, 11, n) − 2M (9, 11, n) ω ) ω ω n − Mω(10, 11, n) q (mod 11). Thanks to Ramanujan’s celebrated congruences (1.1), (1.2) and (1.3), we have proved Corollaries 1.5, 1.6 and 1.7. Acknowledgements. I would like to thank George Andrews for introducing this problem and sharing the manuscript of [3]. Part of this work was ﬁnished during my attendance to the Joint Mathematics Meetings 2019 at Baltimore. References 1. G. E. Andrews, The theory of partitions, Reprint of the 1976 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. xvi+255 pp. 2. G. E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008), 133–142. 3. G. E. Andrews, The Ramanujan–Dyson identities and George Beck’s congruence conjectures, preprint. 4. G. E. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167–171. 5. A. O. L. Atkin and F. G. Garvan, Relations between the ranks and cranks of partitions, Ramanujan J. 7 (2003), no. 1-3, 343–366. 6. A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106. 7. F. J. Dyson, Some guesses in the theory of partitions, Eureka 8 (1944), 10–15. 8. F. J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), no. 2, 169–180. 9. F. G. Garvan, Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank, Int. J. Number Theory 6 (2010), no. 2, 281–309.

Department of Mathematics, Penn State University, University Park, PA 16802, USA E-mail address: [email protected]