RAMANUJAN-TYPE CONGRUENCES FOR THREE COLORED FROBENIUS PARTITIONS

JEREMY LOVEJOY

Abstract. Using the theory of modular forms, we show that the three colored Frobenius partition function vanishes modulo some small primes in certain arithmetic progressions.

1. Introduction

Since their discovery by Ramanujan, the partition congruences

p(5n + 4) ≡ 0 (mod 5)
p(7n + 5) ≡ 0 (mod 7)
p(11n + 6) ≡ 0 (mod 11)

and their generalizations have been the subject of much investigation. We establish a similar batch of congruences for cϕ_{3}(n), the number of three colored Frobenius partitions of n. Namely, we show that for every nonnegative integer n,

cϕ_{3}(45n + 23) ≡ 0 (mod 5)  (1)
cϕ_{3}(45n + 41) ≡ 0 (mod 5)  (2)
cϕ_{3}(63n + 50) ≡ 0 (mod 7)  (3)
cϕ_{3}(99n + 95) ≡ 0 (mod 11)  (4)
cϕ_{3}(171n + 50) ≡ 0 (mod 19)  (5)

The congruence (3) was proved by K. Ono [11], whose techniques are adapted here to prove the remaining congruences. In particular, we will build modular forms whose Fourier coefficients are related to cϕ_{3}(n) and whose congruence properties can be verified with a finite computation.

2. Preliminaries

In generalizing the Frobenius symbol for ordinary partitions, G. Andrews [1] introduced the notion of a k-colored Frobenius partition of n, a two-rowed array

( \begin{matrix}a_{1} & a_{2} & ... & a_{m} \\ b_{1} & b_{2} & ... & b_{m} \end{matrix} )  (6)

where ∑ a_{i}+∑ b_{i}+m = n, and where the {a_{i}}_{i=1}^{m} and {b_{i}}_{i=1}^{m} are decreasing sequences of nonnegative integers in k colors arranged in order first according to size and then according to color. Andrews [1] also found the generating function for the number of such partitions cϕ_{k}(n) when k = 3,

∑_{n=0}^{∞} cϕ_{3}(n)q^{n} = \begin{matrix}(q^{6}; q^{6}) & {5 \atop ∞} & (q^{2}; q^{2}) & {5 \atop ∞} \\ (q; q) & {5 \atop ∞} & (q^{3}; q^{3}) & {2 \atop ∞} & (q^{4}; q^{4}) & {2 \atop ∞} & (q^{12}; q^{12}) & {2 \atop ∞} \end{matrix} + \frac{4q(q^{4}; q^{4})_{∞}^{2}(q^{12}; q^{12})_{∞}^{2}}{(q; q)_{∞}^{3}(q^{2}; q^{2})_{∞}(q^{6}; q^{6})_{∞}}

Recall that (q; q)_{∞} is the usual notation for ∏_{n=1}^{∞} (1 − q^{n}). Later, L. Kolitsch [8] proved a related and for our purposes more useful formula,

∑_{n=0}^{∞} \overline{cϕ_{3}}(n)q^{n} := ∑_{n=0}^{∞} (cϕ_{3}(n) − p (\frac{n}{3})) q^{n} = \frac{9q(q^{9}; q^{9})_{∞}^{3}}{(q^{3}; q^{3})_{∞}(q; q)_{∞}^{3}}  (7)

where p(α) is the ordinary partition function with the convention that p(α) = 0 if α ∉ ℤ. Since the product in equation (7) is essentially a product of η functions, where η(z) := q^{1/24}(q; q)_{∞} and q := e^{2πiz}, we can make use of the following fact [6, 9, 10] to construct modular forms related to the generating function for cϕ_{3}(n):

Proposition 1 (Gordon, Hughes, Newman, Ligozat). Let

f(z) = ∏_{1≤δ|N} η^{r_{δ}}(δz)

be a product of eta functions which satisfies the following criteria (i)

∑_{δ|N} δr_{δ} ≡ 0 (mod 24)

(ii)

∑_{δ|N} \frac{N}{δ} r_{δ} ≡ 0 (mod 24)

(iii)

∏_{δ|N} δ^{r_{δ}} ∈ ℚ^{2}

(iv) For each d|N,

∑_{δ|N} \frac{(d, δ)^{2}r_{δ}}{δ} ≥ 0

Then, f(z) ∈ M_{k}(Γ_{0}(N)) if k = \frac{1}{2}∑ r_{δ} is an integer.

Here M_{k}(Γ_{0}(N)) is the ℂ - vector space of holomorphic modular forms of weight k and level N. Conditions (i)-(iii) ensure that f(\frac{az+b}{cz+d}) = (cz + d)^{k}f(z) for all z with Im(z) > 0 and for all integers a, b, c, d such that ad − bc = 1 and N|c. Condition (iv) guarantees that “f is holomorphic on ℚ.” For more on the basic theory of modular forms, see Koblitz [7]. Our arguments will rely heavily on the following two facts about modular forms and their Fourier coefficients:

Proposition 2. [7] Suppose that f(z) ∈ M_{k}(Γ_{0}(N)) with Fourier expansion f(z) = ∑_{n=0}^{∞} a(n)q^{n}.

(i) For any positive integer t|N,

f(z)|U(t) := ∑_{n=0}^{∞} a(tn)q^{n}

is the Fourier expansion of a modular form in M_{k}(Γ_{0}(N)). (ii) For any positive integer t,

f(z)|V (t) := ∑_{n=0}^{∞} a(n)q^{tn}

is the Fourier expansion of a modular form in M_{k}(Γ_{0}(tN)).

Proposition 3. [12] Suppose f(z) = ∑_{n=0}^{∞} a(n)q^{n} ∈ M_{k}(Γ_{0}(N)) satisfies (i) a(n) ∈ ℤ for all n (ii) a(n) ≡ 0 (mod M) for all n ≤ 1 + \frac{kN}{12} ∏_{p|N} (1 + \frac{1}{p})

Then, a(n) ≡ 0 (mod M) for all n.

Proposition 3 is Sturm’s criterion, where the quantity \frac{kN}{12} ∏_{p|N} (1 + \frac{1}{p}) is known as the Sturm bound. The power of the previous two propositions is that, when the Sturm bound is reasonable, a finite computation can verify congruences in certain progressions of Fourier coefficients of modular forms. Although Sturm’s criterion applies only to holomorphic modular forms, it is often possible to reduce conjectured congruences for coefficients of other series ∑ b(n)q^{n} to finite computations by constructng holomorphic modular forms whose Fourier coefficients are closely related to the b(n). Our primary tool will be the following simple but useful fact:

Lemma 4. Suppose that f = ∑_{n=0}^{∞} a(n)q^{n}, g = ∑_{n=0}^{∞} d(mn)q^{mn} with d(0) = 1, and

fg = ∑_{n=0}^{∞} b(n)q^{n}

Then,

(i) If a(mn + r) ≡ 0 (mod M) for 0 ≤ n ≤ B, then b(mn + r) ≡ 0 (mod M) for 0 ≤ n ≤ B. (ii) If b(mn + r) ≡ 0 (mod M) for all n, then a(mn + r) ≡ 0 (mod M) for all n.

Proof: Both facts follow quickly after writing the coefficient of q^{mn+r} from fg in two ways:

b(mn + r) = a(mn + r) + ∑_{k≥1} d(mk)a(m(n − k) + r)

Part (i) is obvious and part (ii) comes from a simple induction argument.

3. The congruences modulo 11 and 19

We begin with the congruence (4).

Theorem 5. For every non-negative integer n we have

cϕ_{3}(99n + 95) ≡ 0 (mod 11)

Proof: Define the η - product g(z) by

g(z) := \frac{9η^{3}(9z)η^{8}(z)η^{16}(99z)η^{3}(297z)}{η(3z)η(11z)}
= q^{103} ∑_{n=0}^{∞} \overline{cϕ_{3}}(n)q^{n} ∏_{n=1}^{∞} (1 − q^{99n})^{16}(1 − q^{297n})^{3} ∏_{n=1}^{∞} \frac{(1 − q^{n})^{11}}{1 − q^{11n}}

≡ q^{103} ∑_{n=0}^{∞} \overline{cϕ_{3}}(n)q^{n} ∏_{n=1}^{∞} (1 − q^{99n})^{16}(1 − q^{297n})^{3} (mod 11)

:= ∑_{n=103}^{∞} r(n)q^{n}

It is easily verified using Proposition 1 that g(z) ∈ M_{14}(Γ_{0}(297)). By Lemma 4, vanishing modulo 11 is shared by r(99n) and \overline{cϕ_{3}}(99n + 95) = cϕ_{3}(99n + 95). Applying Proposition 2 (i),

g(z)|U(99) = ∑_{n=0}^{∞} r(99n)q^{n}

is also in M_{14}(Γ_{0}(297)). By Sturm’s criterion and Lemma 4 (ii), if r(99n) ≡ 0 (mod 11) for all n ≤ 505, then (4) holds for all n. Moreover, by Lemma 4 (i), r(99n) ≡ 0 (mod 11) for all n ≤ 505 if cϕ_{3}(99n + 95) ≡ 0 (mod 11) for all n ≤ 505. This is verified by machine computation and hence cϕ_{3}(99n + 95) ≡ 0 (mod 11) for all n.

The congruence (5) is shown in a similar way.

Theorem 6. For every non-negative integer n,

cϕ_{3}(171n + 50) ≡ 0 (mod 19)

Proof: Define

h(z) := \frac{9η^{3}(9z)η^{16}(z)η^{8}(171z)η^{3}(513z)}{η(3z)η(19z)}
= q^{121} ∑_{n=0}^{∞} \overline{cϕ_{3}}(n)q^{n} ∏_{n=1}^{∞} (1 − q^{171n})^{8}(1 − q^{513n})^{3} ∏_{n=1}^{∞} \frac{(1 − q^{n})^{19}}{1 − q^{19n}}

≡ q^{121} ∑_{n=0}^{∞} \overline{cϕ_{3}}(n)q^{n} ∏_{n=1}^{∞} (1 − q^{171n})^{8}(1 − q^{513n})^{3} (mod 19)

:= ∑_{n=121}^{∞} r(n)q^{n}

By Proposition 1, h(z) is in M_{14}(Γ_{0}(513)) and hence so is

h(z)|U(171) = ∑_{n=0}^{∞} r(171n)q^{n}

Arguing as above, if cϕ_{3}(171n + 50) ≡ 0 (mod 19) for n ≤ 841, then r(171n) ≡ 0 (mod 19) for n ≤ 841, so r(171n) ≡ 0 (mod 19) for all n, and therefore (5) holds for all n. The congruence for cϕ_{3}(171n + 50) has been checked for n ≤ 841.

4. The congruences modulo 5

The congruences for cϕ_{3}(n) modulo 5 are similar to the congruence modulo 7 and distinct from those modulo 11 and 19 in the sense that they are implied by congruences for

∑_{n=0}^{∞} a(n)q^{n} = ∏_{n=1}^{∞} \frac{1}{(1 − q^{n})^{3}(1 − q^{3n})}  (8)

Theorem 7. The coefficients a(n) as defined above satisfy

a(n) ≡ 0 (mod 5) if n ≡ 13, 22, 31, 40 (mod 45)

Proof: Let f(z) be the η-product defined by

f(z) := \frac{η^{13}(45z)η^{3}(135z)η^{7}(z)}{η(3z)η^{2}(5z)}

= q^{41} ∑_{n=0}^{∞} a(n)q^{n} ∏_{n=1}^{∞} (1 − q^{45n})^{13}(1 − q^{135n})^{3}(∏_{n=1}^{∞} \frac{(1 − q^{n})^{5}}{1 − q^{5n}} )^{2}

≡ q^{41} ∑_{n=0}^{∞} a(n)q^{n} ∏_{n=1}^{∞} (1 − q^{45n})^{13}(1 − q^{135n})^{3} (mod 5)

:= ∑_{n=41}^{∞} r(n)q^{n}

One easily verifies using Proposition 1 that f(z) ∈ M_{10}(Γ_{0}(135)). By Proposition 2 the functions f(z)|U(9) and f(z)|U(45) are in M_{10}(Γ_{0}(135)) and hence the function

f(z)|U(9) − f(z)|U(45)|V (5) = ∑ r(9n)q^{n} − ∑ r(45n)q^{5n}
= ∑_{n≢0 (mod 5)} r(9n)q^{n}

is in M_{10}(Γ_{0}(675)). By Sturm’s criterion, if it can be shown that r(9n) ≡ 0 (mod 5) when n ≤ 901 and n ≢0 (mod 5) then r(45n + 9, 18, 27, 36) ≡ 0 (mod 5) for all n. By Lemma 4 (ii), this will imply that a(45n + 13, 22, 31, 40) ≡ 0 (mod 5) for all n. It has been verified by machine computation that a(45n + 13, 22, 31, 40) ≡ 0 (mod 5) when n ≤ 181 and so by Lemma 4 (i), r(45n + 9, 18, 27, 36) ≡ 0 (mod 5) for n ≤ 181 which proves the theorem.

The proof of the congruences modulo 5 relies on the following well-known corollary of Jacobi’s triple product identity:

∑_{n=0}^{∞} (−1)^{n}(2n + 1)q^{\frac{n^{2}+n}{2}} = ∏_{n=1}^{∞} (1 − q^{n})^{3}

Theorem 8. For every non-negative integer n we have

cϕ_{3}(45n + 23) ≡ 0 (mod 5)
cϕ_{3}(45n + 41) ≡ 0 (mod 5)

Proof: With the notation from (8),

∑ \overline{cϕ_{3}}(n)q^{n} = 9 ∑ a(n)q^{n} ∑ (−1)^{n}(2n + 1)q^{1+ \frac{9n^{2}+9n}{2}} ,

and therefore

cϕ_{3}(45n + 23) = \overline{cϕ_{3}}(45n + 23) = 9 ∑_{k≥0} (−1)^{k}(2k + 1)a (45n + 23 − (1 + \frac{9k^{2} + 9k}{2} ))

cϕ_{3}(45n + 41) = \overline{cϕ_{3}}(45n + 41) = 9 ∑_{k≥0} (−1)^{k}(2k + 1)a (45n + 41 − (1 + \frac{9k^{2} + 9k}{2} ))

Modulo 45, 1 + \frac{9k^{2}+9k}{2} is 1, 10, or 28 and therefore 45n + 23 − (1 + \frac{9k^{2}+9k}{2} ) and 45n + 41 − (1 + \frac{9k^{2}+9k}{2} ) are always congruent to 13, 22, 31, 40 (mod 45). Hence, by Theorem 7, in the above sums the right hand side is always 0 (mod 5).

5. Conclusion

There are some natural questions which arise from the study of congruences of this type. First, are there generalizations of the congruences (1) - (5) to powers of 5,7,11, and 19 analogous to the generalizations of Ramanujan’s congruences for the partition function? Second, to what extent can the congruence properties of cϕ_{3}(n) be understood combinatorially by the use of ranks and cranks, as in the study of p(n) mod 5,7, and 11 [2, 3, 4, 5]? This type of argument has been particularly elusive in the study of congruence properties of other objects similar to (6). In fact, Andrews [1] conjectured such a rank, a_{1} − b_{1}, which might explain some congruences for generalized Frobenius partitions, but this was shown to be untrue.

Acknowledgement

Thanks to Ken Ono, George Andrews, and Kevin James for helpful conversations.

References

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