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arXiv:2407.00058v1 [math.NT] 15 Jun 2024
ARITHMETIC PROPERTIES FOR GENERALIZED CUBIC
PARTITIONS AND OVERPARTITIONS MODULO A PRIME
TEWODROS AMDEBERHAN, JAMES A. SELLERS, AND AJIT SINGH
Abstract. A cubic partition is an integer partition wherein the even parts
can appear in two colors. In this paper, we introduce the notion of gen-
eralized cubic partitions and prove a number of new congruences akin
to the classical Ramanujan-type. We emphasize two methods of proofs,
one elementary (relying significantly on functional equations) and the
other based on modular forms. We close by proving analogous results
for generalized overcubic partitions.
1. Introduction
A partition λof a positive integer nis a sequence of positive integers
λ1≥λ2≥ · · · ≥ λrsuch that Pr
i=1λi=n. The values λ1, λ2, . . . , λrare called
the parts of λ. We let p(n) denote the number of partitions of nfor n≥1,
and we define p(0) :=1.As an example, note that the partitions of n=4
are
4,3+1,2+2,2+1+1,1+1+1+1
and this means p(4) =5.
Throughout this work, we adopt the notations (a;q)∞=Qj≥0(1 −aq j),
where |q|<1, and fk:=(qk;qk)∞. As was proven by Euler, we know that
P(q) :=X
n≥0
p(n)qn=Y
j≥1
1
1−qj=1
f1
.
Because the focus of this paper is on divisibility properties of certain parti-
tion functions, we remind the reader of the celebrated Ramanujan congru-
ences for p(n) [10, p. 210, p. 230]: For all n≥0,
p(5n+4) ≡0 (mod 5),
p(7n+5) ≡0 (mod 7),and(1.1)
p(11n+6) ≡0 (mod 11).
Date: July 2, 2024.
2020 Mathematics Subject Classification. 05A17, 11F03, 11P83.
Key words and phrases. cubic partitions, Ramanujan congruences, modular forms.
1
2 T. AMDEBERHAN, J. A. SELLERS, AND A. SINGH
Indeed, among Ramanujan’s discoveries, the equality
(1.2) X
n≥0
p(5n+4) qn=5(q5;q5)5
∞
(q;q)6
∞
was regarded as his “Most Beautiful Identity” by both Hardy and MacMa-
hon (see [10, p. xxxv]).
Based on an identity on Ramanujan’s cubic continued fractions [1], Chan
[3] introduced the notion of cubic partitions. A cubic partition of weight n
is a partition of nwherein the even parts can appear in two colors. We
denote the number of such cubic partitions by a2(n) and define a2(0) :=
1. For example, a2(3) =4 since the list of cubic partitions of n=3 is:
3,2+1,2+1,1+1+1. We also compute a2(4) =9 because the cubic
partitions of 4 are
4,4,3+1,2+2,2+2,2+2,2+1+1,2+1+1,1+1+1+1.
It is clear from the definition of cubic partitions that the generating function
for a2(n) is given by
F2(q) :=X
n≥0
a2(n)qn=Y
j≥1
1
(1 −qj)(1 −q2j)=1
f1f2
.
Chan [3] succeeded in proving the following elegant analogue of (1.2):
X
n≥0
a2(3n+2) qn=3(q3;q3)3
∞(q6;q6)3
∞
(q;q)4
∞(q2;q2)4
∞
.
It is immediate that, for all n≥0,
(1.3) a2(3n+2) ≡0 (mod 3),
a result similar in nature to (1.1). Since then, many authors have studied
similar congruences for a2(n) (see [5], [6] and references therein). In par-
ticular, we highlight the following result of Chan and Toh [4] which was
proven using modular forms:
Theorem 1.1. For all n ≥0,
a2(5jn+dj)≡0 (mod 5⌊j/2⌋)
where djis the inverse of 8 modulo 5j.
By way of generalization, we define a generalized cubic partition of
weight nto be a partition of n≥1 wherein each even part may appear
in c≥1 different colors. We denote the number of such generalized cu-
bic partitions by ac(n) and define ac(0) :=1. Notice that, for all n≥0,
ARITHMETIC PROPERTIES FOR GENERALIZED CUBIC PARTITIONS 3
a1(n)=p(n) while a2(n) enumerates the cubic partitions of nas described
above. It is clear that the generating function for ac(n) is given by
Fc(q) :=X
n≥0
ac(n)qn=Y
j≥1
1
(1 −qj)(1 −q2j)c−1=1
f1fc−1
2
.(1.4)
Our primary goal in this work is to prove several congruences modulo
primes satisfied by ac(n) for an infinite family of values of c. The proof
techniques that we will utilize will include elementary approaches as well
as modular forms. In particular, we will prove the following infinite family
of congruences modulo a prime p≥3 which are reminiscent of (1.3):
Theorem 1.2. Let p be an odd prime. Then, for all n ≥0,
ap−1(pn +r)≡0 (mod p)
where r is an integer, 1≤r≤p−1, such that 8r+1is a quadratic nonresidue
modulo p.
We will then show that Theorem 1.2 can be generalized in the following
natural way:
Corollary 1.3. Let p ≥3be prime and let r, 1≤r≤p−1such that, for
all n ≥0,ap−1(pn +r)≡0 (mod p). Then, for any k ≥1, and for all n ≥0,
akp−1(pn +r)≡0 (mod p).
We will also prove two “isolated” congruences, modulo the primes 7 and
11, respectively.
Theorem 1.4. For all n ≥0,
a3(7n+4) ≡0 (mod 7) and
a5(11n+10) ≡0 (mod 11).
These two results will be proved via modular forms (using arguments that
are very similar to one another).
The remainder of this paper is organized as follows. In Section 2 we
present elementary proofs of Theorem 1.2 and Corollary 1.3. Section 3 con-
tains the basic results from the theory of modular forms which we need to
prove Theorem 1.4. We close that section with the proof of Theorem 1.4. Fi-
nally, in Section 4, we consider generalized cubic overpartitions and prove
an infinite family of congruences for the related partition functions via ele-
mentary methods. We conclude Section 4 with some relevant remarks.
2. Elementary Proofs of Theorem 1.2 and Corollary 1.3
Sellers [11, Theorem 2.1] developed, among several other results, a func-
tional equation for F2(q). That result can be generalized to our family of
4 T. AMDEBERHAN, J. A. SELLERS, AND A. SINGH
generating functions. Namely, thanks to (1.4), we can easily show that
(2.1) Fc(q)=ψ(q)ψ(q2)c−1Fc(q2)2
where
ψ(q) :=f2
2
f1
=X
k≥0
qk(k+1)/2
is one of Ramanujan’s well–known theta functions [2, p. 6]. We can then
iterate (2.1) to prove the following:
Lemma 2.1. Let p ≥3be prime. Then
Fp−1(q)=ψ(q)Y
i≥1
ψ(q2i)p·2i−1.
Proof. We have
Fp−1(q)=1
f1fp−2
2
=ψ(q)ψ(q2)p−2Fp−1(q2)2
=ψ(q)ψ(q2)p−2ψ(q2)ψ(q4)p−2Fp−1(q4)22
=ψ(q)ψ(q2)pψ(q4)2(p−2) Fp−1(q4)4
=ψ(q)ψ(q2)pψ(q4)2(p−2) ψ(q4)ψ(q8)p−2Fp−1(q8)24
=ψ(q)ψ(q2)pψ(q4)2pψ(q8)4(p−2) Fp−1(q8)8
.
.
.
The result follows by continuing to iterate (2.1) indefinitely.
Lemma 2.1 then leads to a straightforward proof of Theorem 1.2.
Proof of Theorem 1.2. Thanks to Lemma 2.1, we know
Fp−1(q)=ψ(q)Y
i≥1
ψ(q2i)p·2i−1≡ψ(q)Y
i≥1
ψ(qp·2i)2i−1(mod p).
Thus, if we wish to focus our attention on ap−1(pn +r) with the conditions
as stated in the theorem, then we only need to consider ψ(q) because
Y
i≥1
ψ(qp·2i)2i−1
is a function of qp.Thus, we simply need to confirm that we have no solu-
tions to the equation
pn +r=k(k+1)
2
or
8(pn +r)+1=8 k(k+1)
2!+1=(2k+1)2.
ARITHMETIC PROPERTIES FOR GENERALIZED CUBIC PARTITIONS 5
If we did have a solution to the above, then
8r+1≡(2k+1)2(mod p),
but we have explicitly assumed that rhas been selected so that 8r+1 is a
quadratic nonresidue modulo p. Therefore, there can be no such solutions,
and this proves our result.
Proof of Corollary 1.3. The generating function for akp−1(n) is
1
f1fkp−2
2
=1
f1fp−2
2f(k−1)p
2
≡1
f1fp−2
2fk−1
2p
(mod p)
If we wish to focus our attention on ak p−1(pn +r) with the conditions as
stated in the corollary, then we only need to look at 1
f1fp−2
2
because fk−1
2pis
a function of qp. Notice that 1
f1fp−2
2
is the generating function for ap−1(n).
Given that a congruence mod pis assumed to hold for ap−1(n), it must be
the case that a similar congruence modulo pwill hold for ak p−1(n) on the
same arithmetic progression.
3. A Modular Forms Proof of Theorem 1.4
We begin this section with some definitions and basic facts on modular
forms that are instrumental in furnishing our proof of Theorem 1.4. For
additional details, see for example [8, 9]. We first identify the matrix groups
SL2(Z) :=("a b
c d#:a,b,c,d∈Z,ad −bc =1),
Γ0(N) :=("a b
c d#∈SL2(Z) : c≡0 (mod N)),
Γ1(N) :=("a b
c d#∈Γ0(N) : a≡d≡1 (mod N)),
and
Γ(N) :=("a b
c d#∈SL2(Z) : a≡d≡1 (mod N),and b≡c≡0 (mod N)),
where Nis a positive integer. A subgroup Γof the group SL2(Z) is called
acongruence subgroup if Γ(N)⊆Γfor some N. The smallest Nsuch
that Γ(N)⊆Γis called the level of Γ. For example, Γ0(N) and Γ1(N) are
congruence subgroups of level N.
6 T. AMDEBERHAN, J. A. SELLERS, AND A. SINGH
Let H:={z∈C: Im(z)>0}be the upper half of the complex plane. Then,
the following subgroup of the general linear group
GL+
2(R)=("a b
c d#:a,b,c,d∈Rand ad −bc >0)
acts on Hby "a b
c d#z=az +b
cz +d. We identify ∞with 1
0and define
"a b
c d#r
s
=ar +bs
cr +d s ,
where r
s∈Q∪ {∞}. This gives an action of GL+
2(R) on the extended upper
half-plane H∗=H∪Q∪ {∞}. Suppose that Γis a congruence subgroup of
SL2(Z). A cusp of Γis an equivalence class in P1=Q∪{∞} under the action
of Γ. The group GL+
2(R) also acts on functions f:H→C. In particular,
suppose that γ="a b
c d#∈GL+
2(R). If f(z) is a meromorphic function on H
and ℓis an integer, then define the slash operator |ℓby
(f|ℓγ)(z) :=(det γ)ℓ/2(cz +d)−ℓf(γz).
Definition 3.1. Let Γbe a congruence subgroup of level N. A holomorphic
function f:H→Cis called a modular form with integer weight ℓon Γif
the following hold:
(1) We have f(γz)=(cz +d)ℓf(z) for all z∈Hand all γ="a b
c d#∈Γ.
(2) If γ∈SL2(Z), then ( f|ℓγ)(z) has a Fourier expansion of the form
(f|ℓγ)(z)=X
n≥0
aγ(n)qn
N,
where qN:=e2πiz/N.
For a positive integer ℓ, the complex vector space of modular forms of
weight ℓwith respect to a congruence subgroup Γis denoted by Mℓ(Γ).
Definition 3.2. [9, Definition 1.15] If χis a Dirichlet character modulo N,
then we say that a modular form f∈Mℓ(Γ1(N)) has Nebentypus character
χif
f az +b
cz +d!=χ(d)(cz +d)ℓf(z)
for all z∈Hand all "a b
c d#∈Γ0(N). The space of such modular forms is
denoted by Mℓ(Γ0(N), χ).
ARITHMETIC PROPERTIES FOR GENERALIZED CUBIC PARTITIONS 7
In this paper, the relevant modular forms are those that arise from eta-
quotients. The Dedekind eta-function η(z) is defined by
η(z) :=q1/24(q;q)∞=q1/24
∞
Y
n=1
(1 −qn),
where q:=e2πiz and z∈H, the upper half-plane. A function f(z) is called
an eta-quotient if it is of the form
f(z)=Y
δ|N
η(δz)rδ,
where Nis a positive integer and rδis an integer. We now recall two valuable
theorems from [9, p. 18] which will be used to prove our results.
Theorem 3.3. [9, Theorem 1.64] If f (z)=Qδ|Nη(δz)rδis an eta-quotient
such that ℓ=1
2Pδ|Nrδ∈Z,
X
δ|N
δrδ≡0 (mod 24) and X
δ|N
N
δrδ≡0 (mod 24),
then f (z)satisfies f (γz)=χ(d)(cz +d)ℓf(z)for every γ="a b
c d#∈Γ0(N).
Here the character χis defined by χ(•) :=(−1)ℓs
•, where s :=Qδ|Nδrδ.
Suppose that fis an eta-quotient satisfying the conditions of Theorem 3.3
and that the associated weight ℓis a positive integer. If the function f(z)
is holomorphic at all of the cusps of Γ0(N), then f(z)∈Mℓ(Γ0(N), χ). The
next theorem gives a necessary condition for determining orders of an eta-
quotient at cusps.
Theorem 3.4. [9, Theorem 1.65] Let c,d and N be positive integers with
d|N and gcd(c,d)=1. If f is an eta-quotient satisfying the conditions of
Theorem 3.3 for N , then the order of vanishing of f (z)at the cusp c
dis
N
24 X
δ|N
gcd(d, δ)2rδ
gcd(d,N
d)dδ.
We now remind ourselves a result of Sturm [12] which gives a criterion to
test whether two modular forms are congruent modulo a given prime.
Theorem 3.5. Let k be an integer and g(z)=P∞
n=0a(n)qna modular form of
weight k for Γ0(N). For any given positive integer m, if a(n)≡0 (mod m)
holds for all n ≤kN
12 Y
p prime
p|N
1+1
p!,then a(n)≡0 (mod m)holds for any
n≥0.
8 T. AMDEBERHAN, J. A. SELLERS, AND A. SINGH
We now recall the definition of Hecke operators. Let mbe a positive integer
and f(z)=P∞
n=0a(n)qn∈Mℓ(Γ0(N), χ). Then the action of Hecke operator
Tmon f(z) is defined by
f(z)|Tm:=
∞
X
n=0
X
d|gcd(n,m)
χ(d)dℓ−1anm
d2
qn.
In particular, if m=pis prime, we have
f(z)|Tp:=
∞
X
n=0 a(pn)+χ(p)pℓ−1a n
p!!qn.(3.1)
We take by convention that a(n/p)=0 whenever p∤n. The next result
follows directly from (3.1).
Proposition 3.6. Let pbe a prime, g(z)∈Z[[q]],h(z)∈Z[[qp]], and k>1.
Then, we have
(g(z)h(z))|Tp≡g(z)|Tp·h(z/p)(mod p).
With the above in place, we are ready to supply the proof of Theorem 1.4.
Proof of Theorem 1.4. We begin by proving the mod 7 congruence that ap-
pears in the theorem. By taking c=3 in (1.4), we have
X
n≥0
a3(n)qn=Y
k≥1
1
(1 −qk)(1 −q2k)2.(3.2)
Let
H(z) :=η76(z)
η2(2z).
By Theorems 3.3 and 3.4, we find that H(z) is a modular form of weight
37, level 8 and character χ1=(−2−2
•). By (3.2), the Fourier expansion of our
form satisfies
H(z)=
∞
X
n=0
a3(n)qn+3
Y
k≥1
(1 −qk)77.
Using Proposition 3.6, we calculate that
H(z)|T7≡
∞
X
n=0
a3(7n+4)qn+1
Y
k≥1
(1 −qk)11 (mod 7).
Since the Hecke operator is an endomorphism on M37 (Γ0(8), χ1), we have
that H(z)|T7∈M37 (Γ0(8), χ1). By Theorem 3.5, the Sturm bound for this
space of forms is 37. Using S ageMath, we verify that the Fourier coef-
ficients of H(z)|T7up to the desired bound are congruent to 0 modulo 7.
Hence, Theorem 3.5 confirms that H(z)|T7≡0 (mod 7). This completes
the proof of the mod 7 congruence.
ARITHMETIC PROPERTIES FOR GENERALIZED CUBIC PARTITIONS 9
We next consider the mod 11 congruence in the theorem whose proof is
rather similar to the proof of the mod 7 congruence above. By (1.4), we
have
X
n≥0
a5(n)qn=Y
k≥1
1
(1 −qk)(1 −q2k)4.(3.3)
Let
G(z) :=η32(z)
η4(2z).
By Theorems 3.3 and 3.4, we find that G(z) is a modular form of weight
14, level 4 and character χ0=(2−4
•). By (3.3), the Fourier expansion of our
form satisfies
G(z)=
∞
X
n=0
a5(n)qn+1
Y
k≥1
(1 −qk)33.
Using Proposition 3.6, we calculate that
G(z)|T11 ≡
∞
X
n=0
a5(11n+10)qn+1
Y
k≥1
(1 −qk)3(mod 11).
Since the Hecke operator is an endomorphism on M14 (Γ0(4), χ0), we have
that G(z)|T11 ∈M14 (Γ0(4), χ0). By Theorem 3.5, the Sturm bound for
this space of forms is 7. Hence, Theorem 3.5 confirms that G(z)|T11 ≡0
(mod 11). This completes the proof of the mod 11 congruence.
4. Closing Thoughts
We now transition to a consideration of combinatorial objects related to
generalized cubic partitions, namely, generalized cubic overpartitions.
An overpartition of nis a partition of nin which the first occurrence of
a part may be overlined. In 2010, Kim [7] introduced an overpartition ver-
sion of cubic partitions, sometimes called overcubic partitions. Following
the four cubic partitions of n=3 from the Introduction, we find that the
corresponding 12 overcubic partitions of n=3 are given by the following:
3,3,2+1,2+1,2+1,2+1,
2+1,2+1,2+1,2+1,1+1+1,1+1+1.
If we now generalize these to mirror our generalization of cubic partitions,
then the generating function for these cubic overpartitions is given by
Fc(q) :=X
n≥0
ac(n)qn=(−q;q)∞(−q2;q2)c−1
∞
(q;q)∞(q2;q2)c−1
∞
.
We now prove the following natural companion to Corollary 1.3.
10 T. AMDEBERHAN, J. A. SELLERS, AND A. SINGH
Theorem 4.1. Let p ≥3be prime, and let r, 1≤r≤p−1, such that r is
a quadratic nonresidue modulo p. Also let k be a positive integer. Then, for
all n ≥0,ak p−1(pn +r)≡0 (mod p).
Proof. After elementary manipulations, the generating function for ac(n)
can be given in the form
Fc(q)=fc−1
4
f2
1f2c−3
2
.
Now consider the above generating function for c=kp −1. Then we have
X
n≥0
akp−1(n)qn=fk p−2
4
f2
1f2kp−5
2
=
fkp
4
f2kp
2
f5
2
f2
1f2
4!=
fkp
4
f2kp
2
·ϕ(q)
≡
fk
4p
f2k
2p
·ϕ(q) (mod p)
where
ϕ(q) :=1+2X
j≥1
qj2
is another of Ramanujan’s well–known theta functions [2, p. 6]. Note that
fk
4p
f2k
2p
is a function of qp, which means that, in order to prove this theorem, we
simply need to consider whether pn +r=j2for some rand j. Since ris
assumed to be a quadratic nonresidue modulo p, we know that there can be
no solutions to the equation pn +r=j2. This proves our result.
We close this paper with two sets of brief remarks.
(1) We first return to the generalized cubic partitions and make the fol-
lowing elementary observation. For all c>1 which satisfy c≡1
(mod p) for p=5,7 or 11, it is clear that ac(n) “inherits” the corre-
sponding congruence modulo 5, 7, or 11 as Ramanujan’s (1.1). That
is to say, for all j≥0 and n≥0,
a5j+1(5n+4) ≡0 (mod 5),
a7j+1(7n+5) ≡0 (mod 7),and
a11 j+1(11n+6) ≡0 (mod 11).
(2) As we mentioned above, our emphasis in this paper has been on
congruences modulo a prime psatisfied by ac(n) or ac(n) for par-
ticular values of c. We encourage the interested reader to consider
ARITHMETIC PROPERTIES FOR GENERALIZED CUBIC PARTITIONS 11
identifying and proving additional congruences for these families of
functions modulo powers of a prime akin to Theorem 1.1.
References
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New York (2005). 2
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3
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Department of Math ematics, Tulane University, New Orleans, LA 70118, USA
Email address:tamdeber@tulane.edu
Department of Math ematics and Statistics, University of Minnesota Duluth, Duluth,
MN 55812, USA
Email address:jsellers@d.umn.edu
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA
Email address:ajit18@iitg.ac.in
