CONGRUENCES FOR ANDREWS’ SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON’S RANK

F.G. GARVAN

Abstract. Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujantype congruences for spt(n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson’s rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

1. Introduction

Let spt(n) denote the number of smallest parts in the partitions of n. Below is a list of the partitions of 4 with their corresponding number of smallest parts. We see that spt(4) = 10.

4  1
3 + 1  1
2 + 2  2
2 + 1 + 1  2
1 + 1 + 1 + 1 4

In a recent paper Andrews [3] showed that spt(n) is related to the second rank moment. The rank of a partition [19] is the largest part minus the number of parts. The crank of a partition [4] is the largest part if the partition has no ones, otherwise it is the difference between the number of parts larger than the number of ones and the number of ones. We let N(m, n) denote the number of partitions of n with rank m. For n ≠ 1 we let M(m, n) denote the number of partitions of n with crank m. For n = 1 we define

M(−1, 1) = 1, M(0, 1) = −1, M(1, 1) = 1, and otherwise M(m, 1) = 0.

For k even we define

N_{k}(n) = ∑_{m} m^{k}N(m, n)  (kth rank moment),

M_{k}(n) = ∑_{m} m^{k}M(m, n)  (kth crank moment).

Date: August 14, 2008. 2000 Mathematics Subject Classification. Primary 11P83, 11F11, 11F20, 11F33, 11F37; Secondary 05A17, 11P81. Key words and phrases. Partition congruences, rank, crank, Andrews’ smallest parts partition function. The author was supported in part by NSA Grant H98230-07-1-0011. The original version of this paper was written

July 1, 2007. The main results of this paper were first presented at the Illinois Number Theory Fest, May 17, 2007.

Andrews proved that

(1.1)  spt(n) = n p(n) − \frac{1}{2}N_{2}(n),

where p(n) is the number of partitions of n. Dyson [20] gave a combinatorial proof that

(1.2)  \frac{1}{2}M_{2}(n) = n p(n).

Hence we have

(1.3)  spt(n) = \frac{1}{2}(M_{2}(n) − N_{2}(n)).

We make the following Conjecture 1.1.

M_{k}(n) > N_{k}(n),

for k even, k ≥ 2 and n ≥ 1.

The case k = 2 follows from (1.3). We have checked the conjecture for k ≤ 10 and n ≤ 500.

In [3], Andrews found some surprising congruences for spt(n)

(1.4)  spt(5n + 4) ≡ 0 (mod 5),
(1.5)  spt(7n + 5) ≡ 0 (mod 7),
(1.6)  spt(13n + 6) ≡ 0 (mod 13).

These congruences are reminiscent of Ramanujan’s partition congruences

(1.7)  p(5n + 4) ≡ 0 (mod 5),
(1.8)  p(7n + 5) ≡ 0 (mod 7),
(1.9)  p(11n + 6) ≡ 0 (mod 11).

Andrews’ proof of (1.4), (1.5) depends solely on (1.7), (1.8) and known relations for the rank mod 5 and 7. The proof of (1.6) is more difficult and depends on relations mod 13 for the rank due to O’Brien [25].

In Section 2 we prove the following generalization of (1.4)–(1.6): For t = 5, 7 or 13,

spt(n) ≡ −2(n + \frac{t^{2}−1}{24}) p(n) (mod t),

provided 1 − 24n is not a quadratic residue mod t. The proof uses known relations between rank and crank moments [8].

In Section 3 we prove a number of results on Hecke operators and congruences for modular forms that will be needed in later sections. In Section 4 we study spt(n) (mod 11). By using a known relation between rank moments, crank moments and the 23rd power of the eta-function we find that

∑_{n=0}^{∞} spt(11n + 6)q^{n} ≡ 4 ∏_{n=1}^{∞}(1 − q^{n})^{13} (mod 11).

By using the fact that η(24τ)^{13} is a Hecke eigenform we obtain the congruence

spt(11 · 19^{4} · n + 22006) ≡ 0 (mod 11).

In Section 5 we show that the generating functions for the rank moments N_{2k}(11n + 6) (mod 11) can basically be written in terms of half-integral weight Hecke eigenforms. As a result we are able to find the following explicit congruences for the rank

(1.10)  N(r, 11, 5^{4} · 11 · 19^{4} · n + 4322599) ≡ 0 (mod 11),

(1.11)  N(r, 11, 11^{2} · 19^{4} · n + 172904) ≡ 0 (mod 11),

for all 0 ≤ r ≤ 10. Here N(r, t, n) denotes the number of partitions of n with rank congruent to r mod t. Bringmann and Ono [15] conjectured and Bringmann [12] proved that for each prime ℓ > 3 and m, n, u ∈ ℕ there are infinitely many non-nested arithmetic progressions An + B such that

(1.12)  N(r, ℓ^{m}, An + B) ≡ 0 (mod ℓ^{u}),

for all 0 ≤ r ≤ ℓ^{m} − 1. The congruences (1.10) and (1.11) represent the first nontrivial explicit examples of this result. The analogue of (1.12) for N(r, t, An + B) when (t, 2ℓ) = 1 had been proved earlier by Bringmann and Ono [15]. The crank analogue was proved by Mahlburg [24]. All these results generalize the analogue for p(n) proved by Ono [26]. It is also clear by (1.1) that spt(n) can be written as an integer-linear combination of the rank functions N(r, ℓ^{u}, n) mod ℓ^{u}. Thus Bringmann’s result (1.12) also implies that there are infinitely many non-nested arithmetic progressions An + B such that

(1.13)  spt(An + B) ≡ 0 (mod ℓ^{u}).

In Section 6 we describe algorithms for computing congruences for the generating functions of spt(ℓn+ β_{ℓ}) (mod ℓ) when 24β_{ℓ} ≡ 1 (mod ℓ), and Ramanujan-type congruences for spt(n). For certain small primes ℓ we find that an appropriate form for the generating function of spt(ℓn + β_{ℓ}) is congruent mod ℓ to a half-integer weight Hecke eigenform. This leads to explicit examples of (1.13) which we list below.

spt(11 · 19^{4} · n + 22006) ≡ 0 (mod 11),
spt(7^{4} · 17 · n + 243) ≡ 0 (mod 17),
spt(5^{4} · 19 · n + 99) ≡ 0 (mod 19),
spt(13^{4} · 29 · n + 18583) ≡ 0 (mod 29),
spt(29^{4} · 31 · n + 409532) ≡ 0 (mod 31),
spt(5^{4} · 37 · n + 1349) ≡ 0 (mod 37).

These explicit congruences are reminiscent of congruences for the partition function p(n) found earlier by Atkin [6], [7], and Weaver [32]. The connection with half-integer weight Hecke eigenforms is also analogous to what happens for the partition function. This was exploited by Guo and Ono [23], who showed for small ℓ a connection between ℓ-divisibility results for the orders of Tate-Shafarevich groups of certain Tate twists of Dirichlet motives, and partition congruences. Guo and Ono’s results should extend to spt-congruences.

Recently, Folsom and Ono [21] have proved some amazing congruences for spt(n) mod 2 and 3. These congruences had been observed by the author and others. The methods of the present paper do not apply to Folsom and Ono’s results. In the present paper all congruences have prime modulus ℓ where ℓ > 3.

Recently, Bringmann [13] has shown how the generating function for the second rank moment gives rise to a weak Maass form of weight 3/2. This leads to an asymptotic formula and congruences, which in turn also implies the result (1.13).

In the paper [14] we further explore the general problem of congruences mod ℓ for rank moments, Andrews’ [2] symmetrized rank moments and full rank functions for k-marked Durfee symbols for general ℓ. In Section 7 we will present a preview of some of these results and make some concluding remarks.

4 F.G. GARVAN

2. Congruences for spt(n) mod 5, 7 and 13

The following theorem is an extension of Andrews’ congruences (1.4), (1.5), and (1.6). Theorem 2.1. Let t = 5, 7 or 13. Then

(2.1)  spt(n) ≡ −2(n + \frac{t^{2}−1}{24}) p(n) (mod t),

provided 1 − 24n is not a quadratic residue mod t. Proof. We proceed by considering the three cases t = 5, 7, 13 separately. In each case we find the result by reducing a known exact relation between rank and crank moments [8] mod t. t = 5. We need [8, (5.7),p.359]:

N_{6}(n) = \frac{2}{33}(324n^{2} + 69n − 10)M_{2}(n) + \frac{20}{33}(−45n + 4)M_{4}(n) + \frac{18}{11}M_{6}(n)
+ (108n^{2} − 24n + 1)N_{2}(n).

Reducing mod 5, we obtain

(2.2)  N_{6}(n) ≡ (n^{2} + n)M_{2}(n) + 3M_{6}(n) + (3n^{2} + n + 1)N_{2}(n) (mod 5).

Since m^{6} ≡ m^{2} (mod 5), we have

N_{6}(n) ≡ N_{2}(n) (mod 5), M_{6}(n) ≡ M_{2}(n) (mod 5).

Thus we may rewrite (2.2) as

2n(n + 2)N_{2}(n) ≡ (n + 2)(n + 4)M_{2}(n) (mod 5)
≡ 2n(n + 2)(n + 4)p(n) (mod 5) (by (1.2)),

and we have

N_{2}(n) ≡ (n + 4)p(n) (mod 5) if n ≢ 0, 3 (mod 5).

Thus

spt(n) = np(n) − \frac{1}{2}N_{2}(n)
≡ 3(n + 1)p(n) (mod 5) if n ≢ 0, 3 (mod 5),

which gives Theorem 2.1 for the case t = 5. t = 7. This time we need [8, (5.8),p.360] which is a relation between the moments M_{2}, M_{4}, M_{6}, M_{8}, N_{2} and N_{8}. Upon reducing this relation mod 7 and using the fact that

N_{8}(n) ≡ N_{2}(n) (mod 7), M_{8}(n) ≡ M_{2}(n) (mod 7),

we find that

3n(n + 1)(n + 5)N_{2}(n) ≡ 2(n + 1)(n + 5)(n + 6)M_{2}(n) (mod 7)
≡ 4n(n + 1)(n + 5)(n + 6)p(n) (mod 7) (by (1.2)),

and

N_{2}(n) ≡ (6n + 1)p(n) (mod 7) if n ≢ 0, 2, 6 (mod 7).

Thus

spt(n) = np(n) − \frac{1}{2}N_{2}(n)
≡ (5n + 3)p(n) (mod 7) if n ≢ 0, 2, 6 (mod 7),

which gives Theorem 2.1 for the case t = 7.

t = 13. For this case we need [8, (5.10),p.360] which is an exact linear relation between the moments

N_{14}, N_{12}, N_{2}, M_{2}, M_{4}, M_{6}, M_{8}, M_{10}, M_{12}, M_{14}.

Reducing this relation mod 13 we find that

N_{14}(n) ≡ (4 + 4n + 12n^{2} + 4n^{3} + 12n^{4} + 5n^{5} + 8n^{6})M_{2}(n)
+ (1 + 6n + 4n^{2} + 2n^{3} + 3n^{4} + 5n^{5} + 6n^{6})N_{2}(n) + 3M_{14}(n) (mod 13).

Using the fact that

N_{14}(n) ≡ N_{2}(n) (mod 13), M_{14}(n) ≡ M_{2}(n) (mod 13),

we find that

7n(n + 1)(n + 2)(n + 5)(n + 9)(n + 12)N_{2}(n)
≡ 8(n + 1)(n + 2)(n + 5)(n + 9)^{2}(n + 12)M_{2}(n) (mod 13)
≡ 3n(n + 1)(n + 2)(n + 5)(n + 9)^{2}(n + 12)p(n) (mod 13) (by (1.2)),

and

N_{2}(n) ≡ (6n + 2)p(n) (mod 13) if n ≢ 0, 1, 4, 8, 11, 12 (mod 13).

Thus

spt(n) = np(n) − \frac{1}{2}N_{2}(n)
≡ (11n + 12)p(n) (mod 13) if n ≢ 0, 1, 4, 8, 11, 12 (mod 13),

which gives Theorem 2.1 for the case t = 13. □

3. Modular Forms and Hecke Operators

Let N be a positive integer and k be a nonnegative integer. We let M_{k}(N) (resp. S_{k}(N)) be the space of entire modular (resp. cusp) forms of weight k with respect to the modular group Γ_{0}(N). If χ is a Dirichlet character mod N we let M_{k}(N, χ) (resp. S_{k}(N, χ)) be the space of entire modular (resp. cusp) forms of weight k and character χ with respect to the modular group Γ_{0}(N). We define half-integral weight modular forms in the sense of Shimura [29]. If N is divisible by 4 we let M_{k+ \frac{1}{2}} (N, χ) (resp. S_{k+ \frac{1}{2}} (N, χ)) be the space of modular (resp. cusp) forms of weight k + \frac{1}{2} and

character χ with respect to the modular group Γ_{0}(N).

Let GL_{2}^{+}(ℝ) denote the group of all real 2 × 2 matrices with positive determinant. GL_{2}^{+}(ℝ) acts on the complex upper half plane H by linear fractional transformations. As in [29] we let G denote the set of ordered pairs (α, ϕ(τ)), where α ∈ GL_{2}^{+}(ℝ) with last row (c d), and ϕ is a holomorphic function on H such that

ϕ^{2}(τ) = s(det α)^{−1/2}(cτ + d),

where |s| = 1. G is a group with multiplication

(α, ϕ(τ))(β, ψ(τ)) = (αβ, ϕ(βτ)ψ(τ)).

For a holomorphic function f : H ⟶ ℂ and ξ = (α, ϕ(τ)) ∈ G we define

f |_{k+ \frac{1}{2}} ξ = f | ξ = ϕ(τ)^{−2k−1}f(ατ).

6 F.G. GARVAN

For a prime ℓ the Hecke operator T_{k,N,χ}(ℓ^{2}) = T (ℓ^{2}) (as defined in [29]) maps M_{k+ \frac{1}{2}} (N, χ) to itself. If f = ∑_{n=0}^{∞} a(n)q^{n} then f | T (ℓ^{2}) = ∑_{n=0}^{∞} c(n)q^{n} where

(3.1)  c(n) = a(ℓ^{2}n) + χ(ℓ)(\frac{(−1)^{k}n}{ℓ})ℓ^{k−1}a(n) + χ(ℓ^{2})ℓ^{2k−1}a(n/ℓ^{2}).

We note the convention that a(x) = 0 if x is not a nonnegative integer. Let

W_{N} = ((\begin{matrix}0 & −1 \\ N & 0 \end{matrix}) , N^{1/4}\sqrt{−iτ}) ∈ G.

Then the Fricke involution is given by f ↦ f |_{k+ \frac{1}{2}} W_{N}

The Dedekind eta-function is defined by

η(τ) = exp(πiτ/12) ∏_{n=1}^{∞}(1 − exp(2πinτ)) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}),

where as usual q := exp(2πiτ). Then for τ ∈ H

(3.2)  η(−1/τ) = \sqrt{−iτ}η(τ)

and η(24τ) is in S_{\frac{1}{2}} (576, χ_{12}) where

χ_{12}(n) = \begin{cases}1 & if n ≡ ±1 (mod 12), \\ −1 & if n ≡ ±5 (mod 12), \\ 0 & otherwise. \end{cases}

Proposition 3.1. Let 1 ≤ r ≤ 23 with (r, 6) = 1, m be an even nonnegative integer, and ℓ be a prime, ℓ > 3. Define

C_{r,m} := \{η^{r}(24τ)F (24τ) : F ∈ M_{m}(1)\} ⊂ S_{\frac{r}{2} +m}(576, χ_{12}).

Then the Hecke operator T (ℓ^{2}) maps C_{r,m} to C_{r,m}. Proof. Suppose

g(τ) = η^{r}(24τ)F (24τ),

where F ∈ M_{m}(1) and ℓ is a prime, ℓ > 3. Then g(τ) ∈ S_{k+ \frac{1}{2}} (576, χ_{12}) where k = m + \frac{r−1}{2}. By [17, (6)] we have

g | W_{576} | T (ℓ^{2}) = g | T (ℓ^{2}) | W_{576},

since (ℓ, 576) = 1 and χ_{12} is a real character. Since F ∈ M_{m}(1) we have

(3.3)  F (−1/τ) = τ^{m}F (τ).

Hence using (3.2) and (3.3) we find that

g | W_{576} = (\sqrt{24}\sqrt{−iτ})^{−2k−1} g(−1/(576τ))
= (\sqrt{24}\sqrt{−iτ})^{−2k−1} η^{r}(−1/(24τ))F (−1/(24τ))
= i^{m}g.

Now let

H(τ) = \frac{g | T (ℓ^{2})}{η^{r}(24τ)}.

Then

(3.4)  H(−1/(576τ)) = (\sqrt{24}\sqrt{−iτ})^{2k+1} \frac{g | T (ℓ^{2}) | W_{576}}{η^{r}(−1/(24τ))}

= (\sqrt{24}\sqrt{−iτ})^{2k+1} \frac{g | W_{576} | T (ℓ^{2})}{(−24iτ)^{r/2}η^{r}(−1/(24τ))}

= i^{m}(−24iτ)^{k+ \frac{1−r}{2}} \frac{g | T (ℓ^{2})}{η^{r}(24τ)}

= (24τ)^{m}H(τ).

We note the exponents of q in the q-expansion of g(τ) are congruent to r (mod 24). Since ℓ^{2} ≡ 1 (mod 24) we see, by (3.1), that the exponents of q in the q-expansion of g | T (ℓ^{2}) are also congruent to r (mod 24). Since

η^{r}(24τ) = q^{r} + ⋯

and η(τ) is nonzero in H we see that

H(τ) = K(24τ),

for some function K(τ) holomorphic on H that satisfies

K(τ + 1) = K(τ).

From (3.4) we have

K(−1/τ) = H(−1/(24τ)) = τ^{m}H(τ/24) = τ^{m}K(τ).

Hence K(τ) ∈ M_{m}(1) and

g | T (ℓ^{2}) = η^{r}(24τ)K(24τ) ∈ C_{r,m}.
□

Corollary 3.2. Let 1 ≤ r ≤ 23 with (r, 6) = 1, and m be an even nonnegative integer. If dim M_{m}(1) = 1 and 0 ≠ F (τ) ∈ M_{m}(1), then the function

g(τ) = η^{r}(24τ)F (24τ)

is a Hecke eigenform for S_{\frac{r}{2} +m}(576, χ_{12}).

We define the slash operator for modular forms of integer weight. Let k ∈ ℤ. For a holomorphic

function f : H ⟶ ℂ and α = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}^{+}(ℝ) we define

f |_{k} α = f | α = (det α)^{\frac{k}{2}} (cτ + d)^{−k}f(ατ).

For m a positive integer we define the operator U(m) by its action on formal power series

(∑_{n=0}^{∞} a(n)q^{n}) | U(m) = ∑_{n=0}^{∞} a(mn)q^{n}.

We will also need the Hecke operator T_{k}(ℓ) = T (ℓ) on M_{k}(1). For a positive integer m the Hecke operator T (m) (as defined in [5]) maps M_{k}(1) and S_{k}(1) to themselves. Let ℓ be prime. If f = ∑_{n=0}^{∞} a(n)q^{n} then f | T (ℓ) = ∑_{n=0}^{∞} c(n)q^{n} where

c(n) = a(ℓn) + ℓ^{k−1}a(n/ℓ).

Hence, if the coefficients a(n) are integers we have

f | T (ℓ) ≡ f | U(ℓ) (mod ℓ^{k−1}).

Following Chua [18] we define

(3.5)  h_{ℓ}(τ) = (η(τ)η(ℓτ))^{ℓ−1} ,

for ℓ > 3 prime. The following proposition is basically an extension of a result of Chua [18]. The F = 1 case follows from Lemmas 2.1 and 2.2 in [18]. Proposition 3.3. Let ℓ > 3 be prime, k be a nonnegative even integer. If F (τ) ∈ M_{k}(1) then

(3.6)  h_{ℓ}(τ)F (τ) | U(ℓ) + (−1)^{(ℓ−1)/2}ℓ^{k+(ℓ−1)/2−1}h_{ℓ}(τ)F (ℓτ) ∈ S_{k+ℓ−1}(1).

Proof. Suppose ℓ > 3 is prime, k is a nonnegative even integer, and F (τ) ∈ M_{k}(1). Chua [18] showed that h_{ℓ}(τ) ∈ M_{ℓ−1}(ℓ). Actually h_{ℓ}(τ) ∈ S_{ℓ−1}(ℓ) since clearly h_{ℓ} is zero at the cusp ∞ and zero at the cusp 0 by (3.8) below. Hence

g_{ℓ}(τ) = h_{ℓ}(τ)F (τ) ∈ S_{k+ℓ−1}(ℓ).

By [10, Lemma 17(iii), p.144],

(3.7)  g_{ℓ}(τ) | U(ℓ) + ℓ^{(k+ℓ−3)/2}g_{ℓ}(τ) | W_{ℓ} ∈ S_{k+ℓ−1}(1),

where

W_{ℓ} = (\begin{matrix}0 & −1 \\ ℓ & 0 \end{matrix}) .

By (3.2) we have

(3.8)  h_{ℓ} (−\frac{1}{ℓτ} ) = (−1)^{(ℓ−1)/2}ℓ^{(ℓ−1)/2}τ^{ℓ−1}h_{ℓ}(τ).

Since F (τ) ∈ M_{k}(1), we have

(3.9)  F (−\frac{1}{ℓτ} ) = ℓ^{k}τ^{k}F (ℓτ).

Now by (3.8) and (3.9) we have

(3.10)  g_{ℓ} |_{k+ℓ−1} W_{ℓ} = ℓ^{−(k+ℓ−1)/2}τ^{−k−ℓ+1}g_{ℓ} (−\frac{1}{ℓτ} ) = (−1)^{(ℓ−1)/2}ℓ^{k/2}h_{ℓ}(τ)F (ℓτ).

The result (3.6) follows from (3.7) and (3.10). □

Let F ∈ M_{k}(1) and suppose the q-expansion of F (τ) has integer coefficients. We call such a form an integral modular form. We define p(F, n) by

∑_{n=0}^{∞} p(F, n)q^{n} = \frac{F (τ)}{∏_{n=1}^{∞}(1 − q^{n})}.

We prove a straightforward generalization of a theorem due to Chua [18, Theorem 1.1]. We note that Chua’s result was extended to prime power moduli by Ahlgren and Boylan [1].

CONGRUENCES FOR THE SMALLEST PARTS FUNCTION AND THE RANK 9

Theorem 3.4. Suppose ℓ > 3 is prime, k is a nonnegative even integer, and F (τ) ∈ M_{k}(1) is an integral modular form. Define 1 ≤ β_{ℓ} ≤ ℓ − 1 such that 24β_{ℓ} ≡ 1 (mod ℓ) and let

r_{ℓ} = \frac{24β_{ℓ} − 1}{ℓ}, ν_{ℓ} = \frac{ℓ^{2} + 24β_{ℓ} − 1}{24ℓ}.

Then

∑_{n=0}^{∞} p(F, ℓn + β_{ℓ})q^{24n+r_{ℓ}} ≡ η^{r_{ℓ}}(24τ)G_{ℓ,F} (24τ) (mod ℓ),

for some integral modular form G_{ℓ,F} (τ) ∈ M_{k+ℓ−1−12ν_{ℓ}}(1). Proof. Suppose ℓ > 3 is prime, k is a nonnegative even integer, and F (τ) ∈ M_{k}(1) is integral. Then

∑_{n=0}^{∞} p(F, n)q^{24n−1} = \frac{F (24τ)}{η(24τ)}

≡ \frac{F (24τ)}{η(24τ)} \frac{η^{ℓ}(24τ)}{η(24ℓτ)} (mod ℓ)

≡ \frac{h_{ℓ}(24τ)F (24τ)}{η^{ℓ}(24ℓτ)} (mod ℓ),

where h_{ℓ}(τ) is defined in (3.5). Applying the U(ℓ) operator to both sides we obtain

∑_{n=0}^{∞} p(F, ℓn + β_{ℓ})q^{24n+r_{ℓ}} ≡ \frac{h_{ℓ}(24τ)F (24τ) | U(ℓ)}{η^{ℓ}(24τ)} (mod ℓ).

By Proposition 3.3 there is a cusp form j_{ℓ}(τ) ∈ M_{k+ℓ−1}(1) such that

(3.11)  j_{ℓ}(τ) = h_{ℓ}(τ)F (τ) | U(ℓ) + (−1)^{(ℓ−1)/2}ℓ^{k+(ℓ−1)/2−1}h_{ℓ}(τ)F (ℓτ)
= (c_{1}q^{(ℓ^{2}−1)/24} + ⋯) | U(ℓ) + O(q^{(ℓ^{2}−1)/24})
= c_{2}q^{ν_{ℓ}} + ⋯

for some constants c_{1}, c_{2}, since for ℓ ≥ 5, ν_{ℓ} ≤ \frac{ℓ^{2}−1}{24}. It follows that

(3.12)  j_{ℓ}(τ) = Δ^{ν_{ℓ}}(τ)G_{ℓ,F} (τ),

for some G_{ℓ,F} (τ) ∈ M_{k+ℓ−1−12ν_{ℓ}}(1). We note that G_{ℓ,F} (τ) = 0, if k + ℓ − 1 − 12ν_{ℓ} < 0. Here as usual

Δ(τ) = η^{24}(τ).

Since ℓ ≥ 5, k + (ℓ − 1)/2 − 1 ≥ 1, and

h_{ℓ}(τ)F (τ) | U(ℓ) ≡ Δ^{ν_{ℓ}}(τ)G_{ℓ,F} (τ) (mod ℓ),

by (3.11) and (3.12). Since (ℓ, 24) = 1 we have

∑_{n=0}^{∞} p(F, ℓn + β_{ℓ})q^{24n+r_{ℓ}} ≡ \frac{Δ^{ν_{ℓ}}(24τ)G_{ℓ,F} (24τ)}{η^{ℓ}(24τ)} ≡ η^{r_{ℓ}}(24τ)G_{ℓ,F} (24τ) (mod ℓ),

since 24ν_{ℓ} − ℓ = r_{ℓ}. □

4. Congruences for spt(n) mod 11

We need some results for the crank mod 11. For each integer t ≥ 2 and 0 ≤ r ≤ t − 1 let M(r, t, n) denote the number of partitions of n with crank congruent to r mod t. Then for t = 5, t = 7, or t = 11

M(r, t, n) = \frac{1}{t} p(n);

for all n satisfying 24n ≡ 1 (mod t). See [4], [22]. These combinatorial results immediately imply Ramanujan’s partition congruences (1.7), (1.8), (1.9). Here we need the t = 11 case

(4.1)  M(r, 11, 11n + 6) = \frac{1}{11} p(11n + 6),  0 ≤ r ≤ 10.

Let

E(q) = ∏_{n=1}^{∞}(1 − q^{n}).

For r ≥ 1 define p_{r}(n) by

∑_{n=0}^{∞} p_{r}(n)q^{n} = E(q)^{r} = ∏_{n=1}^{∞}(1 − q^{n})^{r},

so that

∑_{n=1}^{∞} p_{23}(n − 1)q^{n} = qE(q)^{23} = q ∏_{n=1}^{∞}(1 − q^{n})^{23}.

We need [8, (5.24),pp.362-3] which is an exact linear relation between rank-crank moments and the function p_{23}(n − 1). Upon reducing this relation mod 11 and using the fact that

N_{12}(n) ≡ N_{2}(n) (mod 11), M_{12}(n) ≡ M_{2}(n) (mod 11),

we find that

p_{23}(11n + 5) ≡ 4N_{2}(11n + 6) + 2M_{2}(11n + 6)
+ M_{4}(11n + 6) + M_{6}(11n + 6) + 10M_{8}(11n + 6) (mod 11).

By (4.1) we have

M_{k}(11n + 6) ≡ ∑_{m=1}^{10} m^{k}M(m, 11, 11n + 6) (mod 11)

≡ M(1, 11, 11n + 6) ∑_{m=1}^{10} m^{k} (mod 11)

≡ 0 (mod 11), if k = 2, 4, 6, 8.

Hence

(4.2)  N_{2}(11n + 6) ≡ 3p_{23}(11n + 5) (mod 11)

and

spt(11n + 6) = (11n + 6)p(11n + 6) − \frac{1}{2}N_{2}(11n + 6)
≡ 4p_{23}(11n + 5) (mod 11).

CONGRUENCES FOR THE SMALLEST PARTS FUNCTION AND THE RANK 11

Now by the Children’s Binomial Theorem, we have

∏_{n=1}^{∞}(1 − q^{n})^{23} = ∏_{n=1}^{∞}(1 − q^{n})^{22} ∏_{n=1}^{∞}(1 − q^{n})

≡ ∏_{n=1}^{∞}(1 − q^{11n})^{2} ∏_{n=1}^{∞}(1 − q^{n}) (mod 11).

We need Euler’s pentagonal number theorem

∏_{n=1}^{∞}(1 − q^{n}) = ∑_{n=−∞}^{∞}(−1)^{n}q^{n(3n−1)/2}.

Since n(3n − 1)/2 ≡ 5 (mod 11) if and only if n ≡ 2 (mod 11), we find that

(4.3)  ∑_{n=0}^{∞} p_{23}(11n + 5)q^{n} ≡ ∏_{n=1}^{∞}(1 − q^{n})^{2} ∏_{n=1}^{∞}(1 − q^{11n}) (mod 11)

≡ ∏_{n=1}^{∞}(1 − q^{n})^{13} (mod 11).

Hence

spt(11n + 6) ≡ 4p_{13}(n) (mod 11).

Now let

F_{13}(τ) = η^{13}(24τ) = ∑_{n≥13} b_{13}(n)q^{n} ∈ S_{\frac{13}{2}} (576, χ_{12}),

where

b_{13}(n) = p_{13} (\frac{n − 13}{24}) .

Then by Corollary 3.2, F_{13} is a Hecke eigenform, so

(4.4) F_{13} | T (ℓ^{2}) = λ_{ℓ} F_{13}, (ℓ > 3),

where λ_{ℓ} ∈ ℤ. We look for the smallest eigenvalue which is a multiple of 11. After some calculation we find that

λ_{19} = −2901404 = −2^{2} · 11 · 23 · 47 · 61 ≡ 0 (mod 11).

By (4.4) and (3.1) we have

b_{13}(19^{3}m) = λ_{19} b_{13}(19m) ≡ 0 (mod 11) when (m, 19) = 1.

We want

19^{3}m ≡ 13 (mod 24), and (m, 19) = 1,

and take

m = 24 · 19k + 7, n = 19^{3}(24 · 19k + 7),

so that

\frac{n − 13}{24} = 19^{4}k + 2000.

Hence

p_{13}(19^{4}n + 2000) ≡ 0 (mod 11),

and spt(19^{4} · 11n + 22006) ≡ 0 (mod 11).

5. Explicit Congruences for the rank mod 11

In this section we find explicit congruences for the rank mod 11. From (4.2) and (4.3) we find that the generating function for N_{2}(11n + 6) is congruent to 3E(q)^{13} (mod 11), where E(q) is defined in Section 4. In the following theorem we give congruences mod 11 for other rank moments, N_{2k}(11n+6), in terms of half-integer weight cusp forms.

Following [8] and Ramanujan [27, p.163] we define

Φ_{j} = Φ_{j}(q) = ∑_{n=1}^{∞} \frac{n^{j}q^{n}}{1 − q^{n}} = ∑_{m,n≥1} n^{j}q^{nm} = ∑_{n=1}^{∞} σ_{j}(n)q^{n},

for j ≥ 1 odd and where σ_{j}(n) = ∑_{d|n} d^{j}. For n ≥ 2 even, we define the Eisenstein series

E_{n}(τ) = 1 − \frac{2n}{B_{n}} Φ_{n−1}(q),

where q = exp(2πiτ), ℑτ > 0 and B_{n} is the n-th Bernoulli number. We note that E_{n}(τ) ∈ M_{n}(1) for n ≥ 4. Theorem 5.1.

(5.1)  ∑_{n=0}^{∞} N_{2} (11n + 6) q^{n} ≡ 3E^{13}(q) (mod 11),

(5.2)  ∑_{n=0}^{∞} N_{4} (11n + 6) q^{n} ≡ 7E^{13}(q) (mod 11),

(5.3)  ∑_{n=0}^{∞} N_{6} (11n + 6) q^{n} ≡ E^{13}(q) (4 + E_{4}(τ)) (mod 11),

(5.4)  ∑_{n=0}^{∞} N_{8} (11n + 6) q^{n} ≡ E^{13}(q) (5 + 6E_{4}(τ) + 6E_{6}(τ)) (mod 11),

(5.5)  ∑_{n=0}^{∞} N_{10} (11n + 6) q^{n} ≡ E^{13}(q) (5 + 4E_{4}(τ) + 6E_{6}(τ) + 6E_{4}^{2}(τ)) (mod 11).

It is clear that each rank moment N_{2k}(11n + 6) (mod 11) can be written in terms of the rank functions N(r, 11, 11n+6), (0 ≤ r ≤ 5). These relations may be inverted to find each N(r, 11, 11n+6) in terms of rank moments mod 11. We note that only r ≤ 5 is needed since N(−m, n) = N(m, n) and N(11 − r, 11, n) = N(r, 11, n). Corollary 5.2.

∑_{n=0}^{∞} N (0, 11, 11n + 6) q^{n} ≡ E^{13}(q) (6 + 7E_{4}(τ) + 5E_{6}(τ) + 5E_{4}^{2}(τ)) (mod 11),

∑_{n=0}^{∞} N (1, 11, 11n + 6) q^{n} ≡ E^{13}(q) (9 + 10E_{6}(τ) + 5E_{4}^{2}(τ)) (mod 11),

∑_{n=0}^{∞} N (2, 11, 11n + 6) q^{n} ≡ E^{13}(q) (4 + 3E_{6}(τ) + 5E_{4}^{2}(τ)) (mod 11),

∑_{n=0}^{∞} N (3, 11, 11n + 6) q^{n} ≡ E^{13}(q) (8 + 4E_{4}(τ) + 6E_{6}(τ) + 5E_{4}^{2}(τ)) (mod 11),

∑_{n=0}^{∞} N (4, 11, 11n + 6) q^{n} ≡ E^{13}(q) (2 + 7E_{4}(τ) + 8E_{6}(τ) + 5E_{4}^{2}(τ)) (mod 11),

∑_{n=0}^{∞} N (5, 11, 11n + 6) q^{n} ≡ E^{13}(q) (7 + 2E_{4}(τ) + 9E_{6}(τ) + 5E_{4}^{2}(τ)) (mod 11).

Although the half-integer weight modular forms appearing in the previous theorem have different weights, each one is a Hecke eigenform (in its corresponding space) in view of Corollary 3.2. As a result we find the following rank congruences. Corollary 5.3.

(5.6)  N(r, 11, 5^{4} · 11 · 19^{4} · n + 4322599) ≡ 0 (mod 11),
(5.7)  N(r, 11, 11^{2} · 19^{4} · n + 172904) ≡ 0 (mod 11),

for all 0 ≤ r ≤ 10.

Before we can prove Theorem 5.1 we need to recall some results on rank and crank moments from [8]. We define the moment generating functions.

R_{k}(q) = ∑_{n≥0} N_{k}(n)q^{n},
C_{k}(q) = ∑_{n≥0} M_{k}(n)q^{n},

for k even. Define

P = P (q) = ∏_{n=1}^{∞} \frac{1}{(1 − q^{n})},

for |q| < 1. In [8] we proved the following recurrence for crank moments

C_{2n} = 2 ∑_{j=1}^{n−1} \binom{2n − 1}{2j − 1} Φ_{2j−1} C_{2n−2j} + 2Φ_{2n−1} P,

which implies that there are integers α_{a_{1},a_{2},...,a_{n}} such that

(5.8)  C_{2n} = 2 P ∑_{a_{1}+2a_{2}+···+na_{n}=n} α_{a_{1},a_{2},...,a_{n}}Φ_{1}^{a_{1}}Φ_{3}^{a_{2}} ⋯ Φ_{2n−1}^{a_{n}}.

In [8] we obtained the following identity

∑_{i=0}^{k−1} \binom{2k}{2i} ∑_{\begin{matrix}α+β+γ=2k−2i \\ α, β, γ even ≥ 0 \end{matrix}} \binom{2k − 2i}{α, β, γ} C_{α} C_{β} C_{γ} P^{−2} − 3 (2^{2k−1} − 1) C_{2} (5.9)

= \frac{1}{2}(2k − 1)(2k − 2)R_{2k} + 6 ∑_{i=1}^{k−1} \binom{2k}{2i} (2^{2i−1} − 1) δ_{q}(R_{2k−2i})

+ ∑_{i=1}^{k−1} [\binom{2k}{2i + 2} (2^{2i+1} − 1) − 2^{2i}\binom{2k}{2i + 1} + \binom{2k}{2i}] R_{2k−2i}.

Here δ_{q} is the differential operator

δ_{q} = q \frac{d}{dq} .

Proof of Theorem 5.1. As noted above, the first congruence (5.1) follows from (4.2) and (4.3). To attack (5.2)–(5.5), we first use (5.9) and (5.8) rewriting each Φ_{2j−1} in terms of E_{2}(τ), E_{4}(τ) and E_{6}(τ).

We define the operator U_{11}^{∗} which acts on q-series by

U_{11}^{∗} (∑_{n=0}^{∞} a(n)q^{n}) = ∑_{n=0}^{∞} a(11n + 6)q^{n}.

We find that

(5.10)  U_{11}^{∗} (R_{4}) ≡ U_{11}^{∗} (6R_{2} + P H^{(4)}) (mod 11),
(5.11)  U_{11}^{∗} (R_{6}) ≡ U_{11}^{∗} (5R_{2} + P H^{(6)}) (mod 11),
(5.12)  U_{11}^{∗} (R_{8}) ≡ U_{11}^{∗} (9R_{2} + P H^{(8)}) (mod 11),
(5.13)  U_{11}^{∗} (R_{10}) ≡ U_{11}^{∗} (9R_{2} + Ψ + P H^{(10)}) (mod 11),

where

H^{(4)} = 9 + 5 E_{2} + 10 E_{4} + 9 E_{2}^{2}
H^{(6)} = 2 + 7 E_{2} + 7 E_{4} + 3 E_{2}^{2} + 8 E_{6} + 8 E_{4}E_{2} + 9 E_{2}^{3}
H^{(8)} = 6 E_{2} + E_{2}^{2} + 6 E_{4} + 10 E_{2}^{3} + 4 E_{6} + 4 E_{4}E_{2} + 9 E_{2}^{4} + 5 E_{2}^{2}E_{4}
+ 10 E_{6}E_{2}
H^{(10)} = 8 + 4 E_{2} + 8 E_{2}^{2} + 4 E_{4} + 3 E_{2}^{3} + 10 E_{6} + 10 E_{4}E_{2} + 9 E_{2}^{4} + 5 E_{2}^{2}E_{4}
+ 10 E_{6}E_{2} + 4 E_{2}^{5} + 10 E_{4}E_{6} + 5 E_{2}^{2}E_{6} + 9 E_{2}^{3}E_{4}

and

(5.14)  Ψ = \frac{1}{11}P (1 − E_{4}E_{6}).

We call a function 11-integral if the coefficients in its q-expansion are rational numbers with bounded denominators relatively prime to 11. We note that all functions involved are 11-integral so that each congruence is well-defined. For example, Ψ is 11-integral since

(5.15)  E_{4}E_{6} = E_{10} = 1 − 264 Φ_{9} ≡ 1 (mod 11).

For each m we write

H^{(m)} = H_{0}^{(m)} + H_{2}^{(m)} + H_{4}^{(m)} + H_{6}^{(m)} + H_{8}^{(m)},

CONGRUENCES FOR THE SMALLEST PARTS FUNCTION AND THE RANK 15

where each H_{j}^{(m)} is congruent mod 11 to an 11-integral modular form in M_{κ(m,j)}(1). Here the weight κ = κ(m, j) ≡ j (mod 10), and we have used [28, Theorem 2,p.22]. We note that

E_{2} ≡ E_{12} (mod 11), and E_{10} ≡ 1 (mod 11).

By Theorem 3.4, each U_{11}^{∗}(P H_{j}^{(m)}) is congruent to a function E^{13}(q)G_{m,j}(τ) where G_{m,j}(τ) is a modular form of weight κ(m, j) − 2. If H_{j}^{(m)} is nonzero we prove a congruence for U_{11}^{∗}(P H_{j}^{(m)}) by checking enough coefficients of q^{n} (i.e., n ≤ ⌊\frac{κ(m,j)−2}{12}⌋). This is a standard argument (see [28], [30]). For example, for m = 10 we have

H^{(10)} = H_{0}^{(10)} + H_{2}^{(10)} + H_{4}^{(10)} + H_{6}^{(10)} + H_{8}^{(10)},

where

H_{0}^{(10)} = 8 + 4 E_{2}^{5} + 10 E_{4}E_{6} + 5 E_{2}^{2}E_{6} + 9 E_{2}^{3}E_{4}  (κ = 60),
H_{2}^{(10)} = 4 E_{2}  (κ = 12),
H_{4}^{(10)} = 8 E_{2}^{2} + 4 E_{4}  (κ = 24),
H_{6}^{(10)} = 3 E_{2}^{3} + 10 E_{6} + 10 E_{4}E_{2}  (κ = 36),
H_{8}^{(10)} = 9 E_{2}^{4} + 5 E_{2}^{2}E_{4} + 10 E_{6}E_{2}  (κ = 48).

We prove the following congruences

U_{11}^{∗}(P H_{0}^{(10)}) ≡ 4 E^{13}(q)E_{4}^{2} (mod 11),
U_{11}^{∗}(P H_{2}^{(10)}) ≡ 0 (mod 11),
U_{11}^{∗}(P H_{4}^{(10)}) ≡ 0 (mod 11),
U_{11}^{∗}(P H_{6}^{(10)}) ≡ 4 E^{13}(q)E_{4} (mod 11),
U_{11}^{∗}(P H_{8}^{(10)}) ≡ 6 E^{13}(q)E_{6} (mod 11),

by checking enough terms. Hence

(5.16)  U_{11}^{∗}(P H^{(10)}) ≡ E^{13}(q) (4 E_{4}^{2} + 4 E_{4} + 6 E_{6}) (mod 11).

Similarly, we find that

(5.17)  U_{11}^{∗}(P H^{(4)}) ≡ 0 (mod 11),
(5.18)  U_{11}^{∗}(P H^{(6)}) ≡ E^{13}(q) E_{4} (mod 11),
(5.19)  U_{11}^{∗}(P H^{(8)}) ≡ E^{13}(q) (6 E_{4} + 6 E_{6}) (mod 11).

To prove (5.5) we also need to compute U_{11}^{∗}(Ψ) mod 11. Since

(\frac{E(q)^{11}}{E(q^{11})})^{11} ≡ 1 (mod 11^{2}),

we have

(5.20)  qU_{11}^{∗}(P (1 − E_{4}E_{6})) ≡ \frac{Δ^{5}(1 − E_{4}E_{6}) | U_{11}}{E(q)^{11}} (mod 11^{2}),

by using an argument similar to the one used in the proof of Theorem 3.4. Now Δ^{5} ∈ S_{60}(1) and E_{4}E_{6}Δ^{5} ∈ S_{70}(1). The set

B_{1} = {ΔE_{4}^{12}, Δ^{2}E_{6}^{6}, Δ^{3}E_{6}^{4}, Δ^{4}E_{6}^{2}, Δ^{5}}

is a basis for S_{60}(1) and multiplying each element by E_{4}E_{6} gives a basis B_{2} for S_{70}(1). We have

Δ^{5} | U_{11} ≡ Δ^{5} |_{60} T (11) (mod 11^{2}),
E_{4}E_{6}Δ^{5} | U_{11} ≡ E_{4}E_{6}Δ^{5} |_{70} T (11) (mod 11^{2}).

We express Δ^{5} |_{60} T (11) in terms of B_{1}, E_{4}E_{6}Δ^{5} |_{70} T (11) in terms of B_{2}, and reduce mod 11^{2} to find

Δ^{5}(1 − E_{4}E_{6}) | U_{11} ≡ 11Δ (1 + E_{4}E_{6}) (E_{4}^{12} + 10 Δ^{4} + 6 Δ^{2}E_{6}^{4} + 7 Δ^{3}E_{6}^{2} + 9 Δ E_{6}^{6})
≡ 22E_{4}^{2}Δ (mod 11^{2}),

by using

Δ = \frac{1}{1728}(E_{4}^{3} − E_{6}^{2}),

and (5.15). It follows from (5.20) that

(5.21)  U_{11}^{∗}(Ψ) ≡ 2E^{13}(q)E_{4}^{2} (mod 11).

Finally, (5.5) follows from (5.1), (5.13), (5.16) and (5.21). This completes the proof of Theorem 5.1.

Proof of Corollary 5.3. From Corollary 5.2 we see that for each 0 ≤ r ≤ 10, there are integers a_{r}, b_{r}, c_{r}, and d_{r} such that

(5.22) ∑_{n=0}^{∞} N(r, 11, \frac{1}{24}(11n + 1))q^{n} ≡ η^{13}(24τ)(a_{r} + b_{r}E_{4}(24τ) + c_{r}E_{6}(24τ) + d_{r}E_{4}^{2}(24τ)) (mod 11).

By Corollary 3.2 we note that each of the functions

η^{13}(24τ), η^{13}(24τ)E_{4}(24τ), η^{13}(24τ)E_{6}(24τ), η^{13}(24τ)E_{4}^{2}(24τ),

is a Hecke eigenform in its corresponding space; i.e., S_{w}(576, χ_{12}) for w = \frac{13}{2}, w = \frac{21}{2}, w = \frac{25}{2}, and w = \frac{29}{2} respectively. For each form we find eigenvalues divisible by 11. For the forms η^{13}(24τ), η^{13}(24τ)E_{4}(24τ), and η^{13}(24τ)E_{4}^{2}(24τ) we find that λ_{19} is divisible by 11. For η^{13}(24τ)E_{6}(24τ), we find that the eigenvalue λ_{5} is divisible by 11. From (5.22) and (3.1), it follows that

N(r, 11, \frac{1}{24}(5^{3} · 11 · 19^{3}n + 1)) ≡ 0 (mod 11)

for each r provided that (n, 5) = (n, 19) = 1. We replace n by 5·19·24·n+c where (c, 5) = (c, 19) = 1 and 5^{3} · 11 · 19^{3}c ≡ −1 (mod 24). The smallest such c is c = 11. This gives

N(r, 11, 5^{4} · 11 · 19^{4}n + 4322599) ≡ 0 (mod 11),

which is (5.6). We claim that

(5.23)  η^{13}(24τ)F (24τ) | U_{11} ≡ 0 (mod 11),

for F (τ) = E_{4}(τ), E_{6}(τ) or E_{4}^{2}(τ). We observe that

η^{13}(24τ)F (24τ) ≡ \frac{Δ(24τ)F (24τ)}{η(24 · 11τ)} (mod 11)

so that

η^{13}(24τ)F (24τ) | U_{11} ≡ \frac{Δ(24τ)F (24τ) | U_{11}}{η(24τ)} (mod 11).

Let F = E_{4} so that Δ(τ)E_{4}(τ) ∈ S_{16}(1). We easily find that

Δ(τ)E_{4}(τ) | U_{11} ≡ Δ(τ)E_{4}(τ) | T (11) ≡ 0 (mod 11),

by checking that the first coefficient is divisible by 11, since dim S_{16}(1) = 1 and Δ(τ)E_{4}(τ) | T (11) ∈ S_{16}(1). This proves (5.23) for the case F = E_{4}. The other two cases are analogous. Now (5.23) together with the fact that the eigenvalue λ_{19} for η^{13}(24τ) is divisible by 11 implies that

N(r, 11, \frac{1}{24}(11^{2} · 19^{3}n + 1)) ≡ 0 (mod 11)

for each r, provided that (n, 19) = 1. We replace n by 19·24·n+c where (c, 19) = 1 and 11^{2}·19^{3}c ≡ −1 (mod 24). The smallest such c is c = 5. This gives

N(r, 11, 11^{2} · 19^{4}n + 172904) ≡ 0 (mod 11),

which is (5.7).

6. Explicit Congruences for spt(n)

For ℓ > 3 prime and β ≥ 1, we define

SPT(ℓ, β) = ∑_{n=0}^{∞} spt(ℓn + β)q^{n}.

In Section 4 we obtained an explicit congruence mod 11 for the case (ℓ, β) = (11, 6). In this section we lay out a more general method for finding congruences mod ℓ. We obtain a number of explicit congruences when ℓ ≤ 37 and 24β ≡ 1 (mod ℓ). Theorem 6.1. We have

(6.1)  SPT(5, 4) ≡ 0 (mod 5),
(6.2)  SPT(7, 5) ≡ 0 (mod 7),
(6.3)  SPT(11, 6) ≡ 4E^{13}(q) (mod 11),
(6.4)  SPT(13, 6) ≡ 0 (mod 13),
(6.5)  SPT(17, 5) ≡ 14E^{7}(q)E_{6}(τ) (mod 17),
(6.6)  SPT(19, 4) ≡ 10E^{5}(q)E_{4}^{2}(τ) (mod 19),
(6.7)  SPT(23, 1) ≡ E(q)E_{4}^{3}(τ) + 7qE^{25}(q) (mod 23),
(6.8)  SPT(29, 23) ≡ 17E^{19}(q)E_{6}(τ) (mod 29),
(6.9)  SPT(31, 22) ≡ 30E^{17}(q)E_{4}^{2}(τ) (mod 31),
(6.10)  SPT(37, 17) ≡ 12E^{11}(q)E_{4}^{2}(τ)E_{6}(τ) (mod 37).

Using Theorem 6.1, Proposition 3.1 and Corollary 3.2 we are able to derive a number of explicit congruences for spt(n). Theorem 6.2. We have

(6.11)  spt(11 · 19^{4} · n + 22006) ≡ 0 (mod 11),
(6.12)  spt(7^{4} · 17 · n + 243) ≡ 0 (mod 17),

(6.13)  spt(5^{4} · 19 · n + 99) ≡ 0 (mod 19),
(6.14)  spt(13^{4} · 29 · n + 18583) ≡ 0 (mod 29),
(6.15)  spt(29^{4} · 31 · n + 409532) ≡ 0 (mod 31),
(6.16)  spt(5^{4} · 37 · n + 1349) ≡ 0 (mod 37).

We prove Theorems 6.1 and 6.2 case by case. As usual we let ℓ > 3 be prime and define 1 ≤ β_{ℓ} ≤ ℓ − 1 such that 24β_{ℓ} ≡ 1 (mod ℓ).

We need some results from [8] and [14]. Let n be a positive integer. Define

W_{2n} = Span_{ℚ}{Φ_{1}^{a}Φ_{3}^{b}Φ_{5}^{c} : 1 ≤ a + 2b + 3c ≤ n with a, b, c nonnegative integers}

so that W_{2n} is a vector space of quasimodular forms of bounded weight over ℚ. We now define what we mean by an “ℓ-integral quasimodular form.” We consider functions E_{2}^{a}(z) F_{b}(z), where F_{b}(z) ∈ M_{b}(1), the coefficients in the q-expansion of F_{b}(z) are ℓ-integral and have bounded denominators, and a and b are nonnegative integers. We call such a function an ℓ-integral quasi-modular form of weight 2a + b. Let k be a nonnegative integer. In general, an ℓ-integral quasi-modular form of weight k is a sum of such functions where 2a + b = k. We let X_{2n,ℓ} denote the subset of ℓ-integral quasimodular forms in W_{2n}. By Theorems 7.4 and 7.6 in [14] there is a certain specific polynomial P_{k}(x) ∈ ℤ[x] such that

(6.17)  R_{2k} − P_{k}(δ_{q})R_{2} ∈ P X_{2k,ℓ} ⊂ P W_{2k},

when 2 ≤ k ≤ \frac{ℓ−3}{2} or k = \frac{ℓ+1}{2}. The result for k = \frac{ℓ−1}{2} is analogous except a factor ℓ may occur in some denominators. We need a basis for P W_{2k} in terms of crank moments and cusp forms. Following [8, (5.14)], we let

C_{2k} = {δ_{q}^{m}(C_{2j}) : 1 ≤ j ≤ k, j + m ≤ k} ⊂ P W_{2k}.

We let B_{2k} be a basis for the space of cusp forms S_{2k}(1), and let

S_{2k} = ∪\begin{matrix}1≤j≤k, \\ 0≤m≤k−j \end{matrix}δ_{q}^{m}(P B_{2j}) ⊂ P W_{2k}.

Conjecture 6.3. For k ≥ 1 the set

T_{2k} = C_{2k} ∪ S_{2k}

forms a basis for P W_{2k} over ℚ.

We have confirmed this conjecture for k ≤ 20 which includes the cases we need. We also note that

dim(P W_{2k}) = |T_{2k}|,

using [8, (3.32)] and the fact that

dim M_{2k}(1) = 1 + dim S_{2k}(1).

For ℓ > 3 prime and ϵ ∈ {−1, 0, 1}, we define the operator U_{ϵ,ℓ}^{∗} which acts on q-series by

(6.18)  U_{ϵ,ℓ}^{∗} (∑ a(n)q^{n}) = ∑_{(\frac{1−24n}{ℓ})=ϵ} a(n)q^{n}.

In [14, Corollary 7.5] we gave an elementary proof that, for ℓ > 3 prime and ϵ = −1 or 0, there is a G_{ℓ} ∈ X_{ℓ+1,ℓ} such that

(6.19)  U_{ϵ,ℓ}^{∗} (R_{2}) ≡ U_{ϵ,ℓ}^{∗} (G_{ℓ}P) (mod ℓ).

The proof depends on (6.17) and the fact that R_{ℓ+1} ≡ R_{2} (mod ℓ), as well some elementary properties of the polynomial P_{\frac{ℓ+1}{2}} (x).

Obtaining a congruence mod ℓ for SPT(ℓ, β_{ℓ}). To prove Theorem 6.1 for the prime ℓ we follow a series of computational steps:

Step 1. First we compute

L_{2k} := R_{2k} − P_{k}(δ_{q})R_{2}

in terms of the set T_{2k} for k = \frac{ℓ−1}{2} and k = \frac{ℓ+1}{2}. This step involves the heaviest computation. In each case this calculation was done in MAPLE by computing the coefficients of q^{j} (0 ≤ j ≤ (n + 20) where n = |T_{2k}|) of each function in T_{2k} as well as the function L_{2k}. These coefficients form a (n + 21) × (n + 1) matrix A, where the first n columns correspond to the functions in T_{2k} and the last column corresponds to the function L_{2k}. We used MAPLE to show that dim Null(A) = 1 and to find a basis vector for this nullspace. In each case the last component of this basis vector is nonzero thus giving the function L_{2k} as a linear combination of the functions in T_{2k}, since we know that L_{2k} ∈ P W_{2k} and |T_{2k}| = dim P W_{2k}.

Step 2. For the identity corresponding to k = \frac{ℓ−1}{2}, we find coefficients are ℓ-integral except for a factor ℓ in some denominators. We multiply both sides of the identity by ℓ, apply the U_{0,ℓ}^{∗} operator (see (6.18)) and reduce mod ℓ. This gives an identity for the second crank moment mod ℓ in terms of certain cusp forms times P. Alternatively, this can be computed by using (1.2).

Step 3. For the identity corresponding to k = \frac{ℓ+1}{2} we apply the U_{0,ℓ}^{∗} operator and reduce mod ℓ. This gives an identity for the second rank moment mod ℓ in terms of certain cusp forms times P. See (6.19).

Step 4. Using (1.3) and the identities from Steps 2 and 3 we obtain a congruence mod ℓ for SPT(ℓ, β_{ℓ}) in terms of U_{0,ℓ}^{∗}(P F), where F is a sum of cusp forms of different weights. We use the theory of modular forms mod ℓ and Theorem 3.4 to simplify and obtain the final result. In each case we find that

SPT(ℓ, β_{ℓ}) ≡ E^{r_{ℓ}}(q)F_{ℓ}(τ) (mod ℓ)

for some F_{ℓ}(τ) ∈ M_{\frac{1}{2} (ℓ−r_{ℓ})+1}(1) ∩ ℤ[[q]].

Obtaining an explicit congruence mod ℓ for spt(n). We let

K_{ℓ}(τ) = η^{r_{ℓ}}(24τ)F_{ℓ}(24τ) ∈ S_{\frac{ℓ+2}{2}} (576, χ_{12}),

where F_{ℓ}(τ) was found in Step 4 above. In most cases the function K_{ℓ}(τ) is a Hecke eigenform.

Step 5. Suppose K_{ℓ}(τ) is a Hecke eigenform; i.e., for each prime Q > 3

K_{ℓ}(τ) | T (Q^{2}) = λ_{Q}K_{ℓ}(τ),

for some integer λ_{Q}. Find the first prime Q such the eigenvalue λ_{Q} ≡ 0 (mod ℓ). Now

∑_{n=0}^{∞} spt(ℓn + β_{ℓ})q^{24n+r_{ℓ}} = ∑_{n=0}^{∞} spt (\frac{ℓn + 1}{24}) q^{n} ≡ η^{r_{ℓ}}(24τ)F_{ℓ}(24τ) ≡ K_{ℓ}(τ) (mod ℓ).

It follows from (3.1) that

(6.20)  spt (\frac{ℓQ^{3}n + 1}{24}) ≡ 0 (mod ℓ),

provided that (n, Q) = 1.

Step 6. For the prime Q found in Step 5 find the smallest integer c such that (c, Q) = 1 and c ≡ −ℓQ (mod 24). Then in (6.20) we replace n by 24Qn + c to obtain

(6.21)  spt(ℓQ^{4}n + \frac{1}{24}(ℓQ^{3}c + 1)) ≡ 0 (mod ℓ).

We note that the constant \frac{1}{24}(ℓQ^{3}c + 1) is an integer since ℓ^{2} ≡ Q^{2} ≡ 1 (mod 24) and c ≡ −ℓQ (mod 24).

We now prove Theorems 6.1 and 6.2 case-by-case by following Steps 1–6. All calculations were done using MAPLE. In Section 4 we gave a proof of the ℓ = 11 case. Here we give the proof of the ℓ = 17 case in detail and sketch the other cases.

ℓ = 17. We compute

L_{16} := R_{16} − P_{8}(δ_{q})R_{2}

in terms of the 40 functions in T_{16}.Then we multiply by 17, apply U_{0,17}^{∗} and reduce mod 17 to find

(6.22)  U_{0,17}^{∗} (C_{2}) ≡ U_{0,17}^{∗} (8E_{4}ΔP) (mod 17).

Next we compute

L_{18} := R_{18} − P_{9}(δ_{q})R_{2}

in terms of the 52 functions in T_{18}. Then we apply U_{0,17}^{∗}, reduce mod 17 and use (6.22) to find

U_{0,17}^{∗} (R_{2}) ≡ U_{0,17}^{∗} (8E_{4}ΔP + 11E_{6}ΔP) (mod 17).

Thus we have

SPT(17, 5) = \frac{1}{2}U_{0,17}^{∗} (C_{2} − R_{2}) ≡ U_{0,17}^{∗} (3E_{6}ΔP) (mod 17).

By Theorem 3.4

U_{0,17}^{∗} (3E_{6}ΔP) ≡ E^{7}(q)G(τ) (mod 17),

for some G(τ) ∈ M_{22}(1) ∩ ℤ[[q]]. A finite computation shows that

(6.23)  G(τ) ≡ 14E_{16}(τ)E_{6}(τ) (mod 17).

In fact we need only verify the coefficients of q^{n} on both sides of (6.23) agree mod 17 for n ≤ ⌊\frac{22}{12}⌋ = 1. But, by [28, Theorem 2(i)],

E_{16}(τ) ≡ 1 (mod 17),

and we have

SPT(17, 5) ≡ 14E^{7}(q)E_{16}(τ)E_{6}(τ) ≡ 14E^{7}(q)E_{6}(τ) (mod 17),

which is (6.5). Thus

∑_{n=0}^{∞} spt(17n + 5)q^{24n+7} ≡ 14G_{17}(τ) (mod 17),

where

G_{17}(τ) = η^{7}(24τ)E_{6}(24τ) ∈ S_{\frac{19}{2}} (576, χ_{12}).

Since dim M_{6}(1) = 1, G_{17}(τ) is a Hecke eigenform by Corollary 3.2. The first eigenvalue divisible by 17 is

λ_{7} = −24959264 = −2^{5} · 11 · 17 · 43 · 97.

We want the smallest integer c satisfying (c, 7) = 1 and c ≡ (−17)7 ≡ 1 (mod 24); i.e., c = 1. As in (6.21) we obtain the congruence

spt(7^{4} · 17 · n + 243) ≡ 0 (mod 17),

which is (6.12).

ℓ = 19. We find

U_{0,19}^{∗} (C_{2}) ≡ U_{0,19}^{∗} (5E_{6}ΔP) (mod 19),
U_{0,19}^{∗} (R_{2}) ≡ U_{0,19}^{∗} (5E_{6}ΔP + E_{4}^{2}ΔP ) (mod 19),
SPT(19, 4) = \frac{1}{2}U_{0,19}^{∗} (C_{2} − R_{2}) ≡ U_{0,19}^{∗} (9E_{4}^{2}ΔP ) .

Using Theorem 3.4 we find that

U_{0,19}^{∗} (9E_{4}^{2}ΔP ) ≡ 10E^{5}(q)E_{18}(τ)E_{4}^{2}(τ) ≡ 10E^{5}(q)E_{4}^{2}(τ) (mod 19).

The result (6.6) follows. Thus

∑_{n=0}^{∞} spt(19n + 4)q^{24n+5} ≡ 10G_{19}(τ) (mod 19),

where

G_{19}(τ) = η^{5}(24τ)E_{4}^{2}(24τ) ∈ S_{\frac{21}{2}} (576, χ_{12}).

Since dim M_{8}(1) = 1, G_{19}(τ) is a Hecke eigenform by Corollary 3.2. The first eigenvalue divisible by 19 is λ_{5}, and (6.13) follows.

ℓ = 23. We find

U_{0,23}^{∗} (C_{2}) ≡ U_{0,23}^{∗} (2E_{10}ΔP) (mod 23),
U_{0,23}^{∗} (R_{2}) ≡ U_{0,23}^{∗} (2E_{10}ΔP + 9Δ^{2}P + 21E_{4}^{3}ΔP ) (mod 23),
SPT(23, 1) = \frac{1}{2}U_{0,23}^{∗} (C_{2} − R_{2}) ≡ U_{0,23}^{∗} (7Δ^{2}P + E_{4}^{3}ΔP ) .

By Theorem 3.4,

U_{0,23}^{∗} (7Δ^{2}P + E_{4}^{3}ΔP ) ≡ E(q)E_{22}(τ)(7Δ(τ) + E_{4}^{3}(τ)) ≡ E(q)(7Δ(τ) + E_{4}^{3}(τ)) (mod 23).

The result (6.7) follows. However in this case the function E(q)(7Δ(τ) + E_{4}^{3}(τ)) is not a Hecke eigenform. ℓ = 29. We find

U_{0,29}^{∗} (C_{2}) ≡ U_{0,29}^{∗} (ΔP (2E_{4}^{2} + 20E_{4}^{4} + 11ΔE_{4})) (mod 29),
U_{0,29}^{∗} (R_{2}) ≡ U_{0,29}^{∗} (ΔP (25E_{4}^{2} + 20E_{4}^{4} + 11ΔE_{4} + 5ΔE_{6})) (mod 29),
SPT(29, 23) = \frac{1}{2}U_{0,29}^{∗} (C_{2} − R_{2}) ≡ U_{0,29}^{∗} (ΔP (3E_{4}^{2} + 12ΔE_{6})) (mod 29).

Using Theorem 3.4, we find U_{0,29}^{∗} (E_{4}^{2}ΔP ) ≡ 0 (mod 29) and

U_{0,29}^{∗} (12E_{6}Δ^{2}P ) ≡ 17E^{19}(q)E_{28}(τ)E_{6}(τ) ≡ 17E^{19}(q)E_{6}(τ) (mod 29).

The result (6.8) follows. Thus

∑_{n=0}^{∞} spt(29n + 23)q^{24n+23} ≡ 17G_{29}(τ) (mod 29),

where

G_{29}(τ) = η^{19}(24τ)E_{6}(24τ) ∈ S_{\frac{31}{2}} (576, χ_{12}).

Since dim M_{6}(1) = 1, G_{29}(τ) is a Hecke eigenform by Corollary 3.2. The first eigenvalue divisible by 29 is λ_{13}, and (6.14) follows.

ℓ = 31. We find

U_{0,31}^{∗} (C_{2}) ≡ U_{0,31}^{∗} (2ΔP (3E_{6}Δ + 6E_{6}^{3} + 2E_{6} + 5E_{10})) (mod 31),
U_{0,31}^{∗} (R_{2}) ≡ ΔP (6E_{6}Δ + 29ΔE_{4}^{2} + 12E_{6}^{3} + 2E_{6} + 30E_{10}) (mod 31),
SPT(31, 22) = \frac{1}{2}U_{0,31}^{∗} (C_{2} − R_{2}) ≡ U_{0,31}^{∗} (ΔP (ΔE_{4}^{2} + E_{6} + 21E_{10})) (mod 31).

By using Theorem 3.4, we find that U_{0,31}^{∗} (E_{6}ΔP) ≡ U_{0,31}^{∗} (E_{10}ΔP) ≡ 0 (mod 31), and

U_{0,31}^{∗} (E_{4}^{2}Δ^{2}P ) ≡ 30E^{17}(q)E_{30}(τ)E_{4}^{2}(τ) ≡ 30E^{17}(q)E_{4}^{2}(τ) (mod 31).

The result (6.9) follows. Thus

∑_{n=0}^{∞} spt(31n + 22)q^{24n+17} ≡ 30G_{31}(τ) (mod 31),

where

G_{31}(τ) = η^{17}(24τ)E_{4}^{2}(24τ) ∈ S_{\frac{33}{2}} (576, χ_{12}).

Since dim M_{8}(1) = 1, G_{31}(τ) is a Hecke eigenform by Corollary 3.2. The first eigenvalue divisible by 31 is λ_{29}, and (6.15) follows.

ℓ = 37. We find

U_{0,37}^{∗} (C_{2}) ≡ U_{0,37}^{∗} (ΔP (22Δ^{2} + 4Δ + 31ΔE_{6}^{2} + 17ΔE_{4} + 4E_{4}^{3} + 3E_{10}
+27E_{4}^{4} + 12E_{4}^{6})) (mod 37),
U_{0,37}^{∗} (R_{2}) ≡ U_{0,37}^{∗} (ΔP (22Δ^{2} + 36Δ + 31ΔE_{6}^{2} + 35ΔE_{4} + 20ΔE_{6}E_{4}^{2} + 36E_{4}^{4}
+13E_{14}E_{4}^{3} + 23E_{10} + 36E_{4}^{3} + 12E_{4}^{6})) (mod 37),
SPT(37, 17) = \frac{1}{2}U_{0,37}^{∗} (C_{2} − R_{2})

≡ U_{0,37}^{∗} (ΔP (21Δ + 28ΔE_{4} + 27ΔE_{6}E_{4}^{2} + 27E_{10} + 14E_{4}^{4} + 21E_{4}^{3} + 12E_{14}E_{4}^{3})) (mod 37).

Using Theorem 3.4 we find that

U_{0,37}^{∗} (E_{10}ΔP) ≡ U_{0,37}^{∗} (ΔP (Δ + E_{4}^{3})) ≡ U_{0,37}^{∗} (ΔP (28ΔE_{4} + 14E_{4}^{4})) ≡ 0 (mod 37),

and

U_{0,37}^{∗} (ΔP (27ΔE_{6}E_{4}^{2} + 12E_{14}E_{4}^{3})) ≡ 12E^{11}(q)E_{36}(τ)E_{4}^{2}(τ)E_{6}(τ) ≡ 12E^{11}(q)E_{4}^{2}(τ)E_{6}(τ) (mod 37).

The result (6.10) follows. Thus

∑_{n=0}^{∞} spt(37n + 17)q^{24n+11} ≡ 12G_{37}(τ) (mod 37),

where

G_{37}(τ) = η^{11}(24τ)E_{4}^{2}(24τ)E_{6}(24τ) ∈ S_{\frac{39}{2}} (576, χ_{12}).

Since dim M_{14}(1) = 1, G_{37}(τ) is a Hecke eigenform by Corollary 3.2. The first eigenvalue divisible by 37 is λ_{5} and (6.16) follows.

7. Concluding remarks

In a companion paper [14] we further explore the general problem of congruences mod ℓ for rank moments, Andrews’ [2] symmetrized rank moments and full rank functions for k-marked Durfee symbols for general ℓ. Let ℓ > 3 be prime, and β_{ℓ}, r_{ℓ} be as in Theorem 3.4. By a detailed ℓ-adic analysis of the rank-crank moment equation (5.9) we have been able to show that

(7.1)  ∑_{n=0}^{∞} N_{2k}(ℓn + β_{ℓ})q^{24n+r_{ℓ}} ≡ η^{r_{ℓ}}(24τ) G_{ℓ,2k}(24τ) (mod ℓ),

where G_{ℓ,2k}(τ) is a sum of level one integral modular forms of bounded weight when 2 ≤ 2k ≤ ℓ − 3. When 2k = ℓ − 1 the result is similar but involves an additional function analogous to Ψ, given in (5.14). By using a result of Treneer [31, Prop. 5.2] it can be shown that there exists a positive proportion of the primes Q for which the appropriate Hecke operator T (Q^{2}) annihlates mod ℓ all the half integer weight forms that occur on the right side of (7.1). This implies that there exists a positive proportion of the primes Q such that

(7.2)  N_{2k}(ℓQ^{4}n + μ) ≡ 0 (mod ℓ),

for all integers k, n ≥ 0. Here μ is an integer constant that depends on ℓ and Q. As a result this leads to infinitely many Ramanujan-type congruences for Dyson’s rank function N(r, ℓ, n) (the number of partitions of n with rank congruent to r mod ℓ). In particular, this implies that for each prime ℓ > 3 there is a positive proportion of the primes Q such that

(7.3)  N(r, ℓ, ℓQ^{4}n + μ) ≡ 0 (mod ℓ),

for all integers n ≥ 0 and 0 ≤ r ≤ ℓ − 1. Again μ is an integer constant that depends on ℓ and Q. This corresponds to Bringmann’s result (1.12) (with m = u = 1) which was proved using the theory of Maass forms. As noted before, since spt(n) involves p(n) and the second rank moment N_{2}(n), this leads to another proof that there are infinitely many Ramanujan-type congruences mod ℓ (ℓ > 3 prime) for spt(n). In particular, this implies that for each prime ℓ > 3 there is a positive proportion of the primes Q such that

(7.4)  spt(ℓQ^{4}n + μ) ≡ 0 (mod ℓ),

for all integers k, n ≥ 0, where μ is an integer constant that depends on ℓ and Q. We note that Bringmann [13] proved this result by a different method.

In a recent paper, Bringmann, Ono and Rhoades [16] have shown that the generating function for many rank differences is a modular form. In particular, they have shown the function

(7.5)  ∑_{n=0}^{∞} (N(r_{1}, ℓ, ℓn + d) − N(r_{2}, ℓ, ℓn + d)) q^{24(ℓn+d)−1}

is a weakly holomorphic weight 1/2 modular form when 1 − 24d is a quadratic nonresidue mod ℓ and when it is a quadratic residue under certain additional conditions on r_{1} and r_{2}. As noted in [16] this was observed earlier by Atkin and Swinnerton-Dyer [11] in the cases ℓ = 5 and ℓ = 7. It should also be noted that it has also been observed when ℓ = 11 by Atkin and Hussain [9], and when ℓ = 13 by O’Brien [25]. What is more interesting is the result also holds in the case 24d ≡ 1 (mod ℓ) for ℓ = 11 and ℓ = 13 (again [9] and [25]). We conjecture that the analogue for the case 24d ≡ 1 (mod ℓ) holds for all primes ℓ > 3. The cases ℓ = 5, 7 are trivial since the rank differences are zero. We can prove that in the case 24d ≡ 1 (mod ℓ) the generating function for the rank differences is congruent mod ℓ to a sum of half-integer weight cusp forms. An analogue of this also holds in the case that 1 − 24d is a quadratic nonresidue mod ℓ. Acknowledgement I would like to thank the following for helpful comments: Kathrin Bringmann, Byungchan Kim, Karl Mahlburg, and Ken Ono. I would also like to thank the referees for a careful reading of the manuscript and their many helpful suggestions.

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Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

   E-mail address: frank@math.ufl.edu