RAMANUJAN-TYPE CONGRUENCES FOR THREE COLOREDFROBENIUS PARTITIONSJEREMY 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 congruencesp(5n + 4) ≡ 0p(7n + 5) ≡ 0p(11n + 6) ≡ 0(mod 5)(mod 7)(mod 11)and their generalizations have been the subject of much investigation. We establish a similarbatch of congruences for cφ3 (n), the number of three colored Frobenius partitions of n.
Namely, we show that for every nonnegative integer n,
cφ3 (45n + 23)
cφ3 (45n + 41)
cφ3 (63n + 50)
cφ3 (99n + 95)
cφ3 (171n + 50)

≡≡
≡
≡
≡

0
00
00

(mod 5)(mod 5)(mod 7)(mod 11)(mod 19)

(1)
(2)
(3)
(4)
(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 cφ3 (n) and whose congruence properties can be verified with a finitecomputation.
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¶µa1 a2 ... am(6)b1 b2 ... bm
PPmwhereai + bi + m = n, and where the {ai }mi=1 and {bi }i=1 are decreasing sequences ofnonnegative integers in k colors arranged in order first according to size and then according
Date: January 21, 2004.
1

2to color. Andrews [1] also found the generating function for the number of such partitionscφk (n) when k = 3,∞
X552(q 6 ; q 6 )∞ (q 2 ; q 2 )∞ncφ3 (n)q =24q(q 4 ; q 4 )∞ (q 12 ; q 12 )∞+(q; q)5∞ (q 3 ; q 3 )2∞ (q 4 ; q 4 )2∞ (q 12 ; q 12 )2∞ (q; q)3∞ (q 2 ; q 2 )∞ (q 6 ; q 6 )∞Qn
Recall that (q; q)∞ is the usual notation for ∞n=1 (1 − q ). Later, L. Kolitsch [8] proved arelated and for our purposes more useful formula,n=0∞
Xn=0∞ ³³ n ´´X9q(q 9 ; q 9 )3∞
qn = 3 3cφ3 (n)q :=cφ3 (n) − p3(q ; q )∞ (q; q)3∞n(7)n=0where p(α) is the ordinary partition function with the convention that p(α) = 0 if α ∈/ Z. Since
1/24
the product in equation (7) is essentially a product of η functions, where η(z) := q(q; q)∞
and q := e2πiz , we can make use of the following fact [6, 9, 10] to construct modular forms
related to the generating function for cφ3 (n):Proposition 1 (Gordon, Hughes, Newman, Ligozat). Let
Yf (z) =η rδ (δz)
1≤δ|N

be a product of eta functions which satisfies the following criteria(i )Xδrδ ≡ 0 (mod 24)

δ|N(ii )XN
δ|N(iii )δrδ ≡ 0 (mod 24)Yδ rδ ∈ Q2

δ|N(iv ) For each d|N ,X (d, δ)2 rδ
δ|Nδ

≥0P
Then, f (z) ∈ Mk (Γ0 (N )) if k = 12 rδ is an integer.
Here Mk (Γ0 (N )) is the C - vector space of holomorphic modular forms of weight k andklevel N . Conditions (i )-(iii ) ensure that f ( az+bcz+d ) = (cz + d) f (z) for all z with Im(z) > 0and for all integers a, b, c, d such that ad − bc = 1 and N |c. Condition (iv ) guarantees that “fis holomorphic on Q.” For more on the basic theory of modular forms, see Koblitz [7]. Ourarguments will rely heavily on the following two facts about modular forms and their Fourier
coefficients:

3

Proposition 2. [7] Suppose that f (z) ∈ Mk (Γ0 (N )) with Fourier expansionf (z) =∞Xa(n)q n .

n=0(i ) For any positive integer t|N ,f (z)|U (t) :=∞Xa(tn)q nn=0

is the Fourier expansion of a modular form in Mk (Γ0 (N )).
(ii ) For any positive integer t,∞
Xf (z)|V (t) :=a(n)q tn

n=0

is the Fourier expansion of a modular form in Mk (Γ0 (tN )).
PnProposition 3. [12] Suppose f (z) = ∞n=0 a(n)q ∈ Mk (Γ0 (N )) satisfies(i ) a(n) ∈ Z for all nQ1(ii ) a(n) ≡ 0 (mod M ) for all n ≤ 1 + kN
p|N (1 + p )12Then, a(n) ≡ 0 (mod M ) for all n.
Q1Proposition 3 is Sturm’s criterion, where the quantity kN
p|N (1 + p ) is known as the12Sturm bound. The power of the previous two propositions is that, when the Sturm bound isreasonable, a finite computation can verify congruences in certain progressions of Fourier coefficients of modular forms. Although Sturm’s criterion applies only to holomorphic modularforms,
it is often possible to reduce conjectured congruences for coefficients of other seriesPb(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:PP∞
nmn with d(0) = 1, andLemma 4. Suppose that f = ∞n=0 a(n)q , g =n=0 d(mn)qfg =∞Xb(n)q nn=0Then,(i ) If a(mn + r) ≡ 0 (mod M ) for 0 ≤ n ≤ B, then b(mn + r) ≡ 0 (mod M ) for0 ≤ 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 f g in two ways:
Xb(mn + r) = a(mn + r) +d(mk)a(m(n − k) + r)
k≥1

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

¤4

3. The congruences modulo 11 and 19
We begin with the congruence (4).
Theorem 5. For every non-negative integer n we havecφ3 (99n + 95) ≡ 0(mod 11)

Proof: Define the η - product g(z) byg(z) :==≡

9η 3 (9z)η 8 (z)η 16 (99z)η 3 (297z)
η(3z)η(11z)
∞
∞
∞
XYY
(1 − q n )11
103n99n 16297n 3q
cφ3 (n)q(1 − q ) (1 − q)
1 − q 11nq 103

n=0
∞
X

cφ3 (n)q nn=0

:=

∞
X

n=1∞
Yn=1

163

(1 − q 99n ) (1 − q 297n )(mod 11)n=1r(n)q nn=103

It is easily verified using Proposition 1 that g(z) ∈ M14 (Γ0 (297)). By Lemma 4, vanishing
modulo 11 is shared by r(99n) and cφ3 (99n + 95) = cφ3 (99n + 95). Applying Proposition 2(i ),∞Xg(z)|U (99) =r(99n)q nn=0is also in M14 (Γ0 (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 cφ3 (99n + 95) ≡ 0 (mod 11) for all n ≤ 505. This is verified by machine
computation and hence cφ3 (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,cφ3 (171n + 50) ≡ 0 (mod 19)
Proof: Defineh(z) :==≡9η 3 (9z)η 16 (z)η 8 (171z)η 3 (513z)
η(3z)η(19z)
∞
∞
∞n 19
X
Y
83 Y (1 − q )(1 − q 171n ) (1 − q 513n )q 121
cφ3 (n)q n
1 − q 19n
q 121

n=0∞
X

cφ3 (n)q nn=0:=∞X
n=121r(n)q nn=1∞Y
n=1n=183(1 − q 171n ) (1 − q 513n )(mod 19)

5By Proposition 1, h(z) is in M14 (Γ0 (513)) and hence so ish(z)|U (171) =∞Xr(171n)q nn=0Arguing as above, if cφ3 (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 cφ3 (171n + 50) has been checked for n ≤ 841.
¤4. The congruences modulo 5
The congruences for cφ3 (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∞Xna(n)q =n=0∞Y

1n=1(1 − q n )3 (1 − q 3n )(8)Theorem 7. The coefficients a(n) as defined above satisfya(n) ≡ 0 (mod 5) if n ≡ 13, 22, 31, 40 (mod 45)
Proof: Let f (z) be the η-product defined byf (z) :==≡

η 13 (45z)η 3 (135z)η 7 (z)η(3z)η 2 (5z)
q 41
q 41∞
Xn=0∞
Xa(n)q na(n)q nn=0:=∞
X∞
Yn=1∞
Y

(1 − q45n 13) (1 − q135n 3)

Ã∞!2Y (1 − q n )5n=1133(1 − q 45n ) (1 − q 135n )1 − q 5n(mod 5)

n=1r(n)q n

n=41

One easily verifies using Proposition 1 that f (z) ∈ M10 (Γ0 (135)). By Proposition 2 thefunctions f (z)|U (9) and f (z)|U (45) are in M10 (Γ0 (135)) and hence the function
XXf (z)|U (9) − f (z)|U (45)|V (5) =r(9n)q n −r(45n)q 5n
X=r(9n)q nn6≡0(mod 5)

is in M10 (Γ0 (675)). By Sturm’s criterion, if it can be shown that r(9n) ≡ 0 (mod 5) whenn ≤ 901 and n 6 ≡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. ¤6The proof of the congruences modulo 5 relies on the following well-known corollary of
Jacobi’s triple product identity:∞∞XYn2 +nn2(−1) (2n + 1)q=(1 − q n )3n=0

n=1

Theorem 8. For every non-negative integer n we havecφ3 (45n + 23) ≡ 0 (mod 5)
cφ3 (45n + 41) ≡ 0 (mod 5)
Proof: With the notation from (8),XXX
9n2 +9ncφ3 (n)q n = 9a(n)q n(−1)n (2n + 1)q 1+ 2 ,and thereforeµµ¶¶
9k 2 + 9kcφ3 (45n + 23) = cφ3 (45n + 23) = 9(−1) (2k + 1)a 45n + 23 − 1 +2k≥0µµ¶¶X
9k 2 + 9kk(−1) (2k + 1)a 45n + 41 − 1 +cφ3 (45n + 41) = cφ3 (45n + 41) = 92Xkk≥0

22Modulo 45, 1 + 9k 2+9k is 1, 10, or 28 and therefore 45n + 23 − (1 + 9k 2+9k ) and 45n + 41 −2(1 + 9k 2+9k ) are always congruent to 13, 22, 31, 40 (mod 45). Hence, by Theorem 7, in theabove 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 cφ3 (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 ofargument has been particularly elusive in the study of congruence properties of other objects
similar to (6). In fact, Andrews [1] conjectured such a rank, a1 − b1 , which might explainsome congruences for generalized Frobenius partitions, but this was shown to be untrue.
Acknowledgement
Thanks are due to Ken Ono, George Andrews, and Kevin James for helpful conversations.
References[1] G.E. Andrews, Generalized Frobenius Partitions, Mem. Amer. Math. Soc. 49 (1984), 301.
[2] G.E. Andrews and F.G. Garvan, Dyson’s Crank of a Partition, Bull. Amer. Math. Soc. 19 (1988), 167-171.
[3] A.O.L. Atkin and P. Swinnerton-Dyer, Some Properties of Partitions, Proc. London Math. Soc. (3) 4(1954), 84-106.
[4] F. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7, and 11,Trans. Amer. Math. Soc. 305 (1988), 47-77.
[5] F. Garvan, D. Kim, and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), 1-17.
[6] B. Gordon and K. Hughes, Multiplicative properties of η-products II, Cont. Math. 143 (1993), 415-430.

7[7] N. Koblitz, ”Introduction to Elliptic Curves and Modular Forms,” Springer-Verlag, Berlin/New York,1984.
[8] L. Kolitsch, A congruence for generalized Frobenius partitions with 3 colors modulo powers of 3, ”Analytic
Number Theory, Proceedings of a Conference in Honor of Paul Bateman,” pp. 343-348, Birkhäuser, Basel,
1990.
[9] G. Ligozat, ”Courbes modulaires de genre 1,” Vol.43, Bulletin de la Soc. Math. de France, 1975.
[10] M. Newman, Construction and application of a certain class of modular functions II, Proc. London Math.
Soc. (3) 9 (1959), 353-387.
[11] K. Ono, Congruences for Frobenius Partitions, J. Number Th. 57 (1996), 170-180.
[12] J. Sturm, On the Congruence Properties of Modular Forms, Springer Lect. Notes in Math. Vol.1240,
pp. 275-280, Springer-Verlag, Berlin/New York, 1984.
Department of Mathemtics,, University of Wisconsin,, University Park, WI 53706
E-mail address: lovejoy@math.psu.edu