A THEOREM ON SEVEN-COLORED OVERPARTITIONS AND ITSAPPLICATIONSJEREMY LOVEJOY
Abstract. A q-series identity in four parameters is established and interpreted it as a statement about 7-colored overpartitions. As corollaries some overpartition theorems of the RogersRamanujan type and some weighted overpartition theorems are exhibited. Among these are
overpartition analogues of classical partition theorems of Schur and Göllnitz.

1. Introduction
An overpartition is a partition in which the first occurrence of a number may be overlined.
It has recently been shown [15, 16] that some of the classical techniques Andrews used to study
Rogers-Ramaujan type identities for ordinary partitions are equally well-suited to the study of
overpartitions. Here we turn our attention to the method of colored partitions that has beendeveloped during the last decade [5, 6, 7] and show that overpartitions have a natural place
there as well.
We begin by establishing the following q-series identity in four parameters, wherein we employ
the common notation
Tm := m(m + 1)/2

(1.1)

and
(a1 , a2 , ..., ak )n := (a1 , a2 , ..., ak ; q)n :=n−1
Y(1 − a1 q j )(1 − a2 q j ) · · · (1 − ak q j ),

(1.2)

j=0

the first definition being valid for all integers m and the second for non-negative integers n.
Lemma 1.1.
(−aq, −bq, −cq)∞=(1.3)(adq, bdq, cdq)∞X
q s+Tδ +T +Tφ−1 +` (1 − q α (1 − q φ ))aα+δ++` bβ+δ+φ+` cγ++φ+` ds+` (−1/d)s (−1)`.
(q)α (q)β (q)γ (q)δ (q) (q)φ (q)`α,β,γ,δ,,φ,`≥0
s=α+β+γ+δ++φ+`Then we show that the identity may be interpreted in terms of certain overpartitions whoseparts occur in seven colors. We denote these seven colors by a, b, c, ab, ac, bc, and abcd, with theconvention that parts are ordered first according to size and then according to color, where thecolors are ordered by
Date: April 19, 2007.
2000 Mathematics Subject Classification. 11P81, 05A17.
1

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JEREMY LOVEJOY

abcd < ab < ac < a < bc < b < c.
(1.4)
The relevant seven-colored overpartitions are defined below. For any part λi , we use ci to denoteits color.
Definition 1.2. Let C(α, β, γ, δ, , φ, `; k; n) denote the number of overpartitions of n occurringin the seven colors above with α a-parts, β b-parts, γ c-parts, δ ab-parts,  ac-parts, φ bc-parts,` abcd-parts, k non-overlined parts, and the following additional properties:(i) 1 cannot occur, overlined or not, in colors ab, ac, or abcd.
(ii) 1 and 1 can occur in color bc only if 1 also occurs in color a.
(iii) λi − λi+1 is at least one if the smaller is overlined OR if ci = ci+1 ∈ {ab, ac, bc} OR if
ci < ci+1 in the order (1.4),(iv) λi − λi+1 is at least two if the smaller is overlined AND ci = ci+1 ∈ {ab, ac, bc} or
ci < ci+1 in the order (1.4).
The principal result in this paper is the following, which asserts that a weighted generating
function for the overpartitions defined above is an infinite product.
Theorem 1.3.
X(−aq, −bq, −cq)∞
C(α, β, γ, δ, , φ, `; k; n)aα+δ++` bβ+δ+φ+` cγ++φ+` dk+` (−1)` =.
(adq, bdq, cdq)∞
α,β,γ,δ,,φ,`,k,n≥0

Our first two applications of Theorem 1.3 are a generalization of a partition theorem of Schur[17] and a result that is very much like a partition theorem of Göllnitz [6, Theorem G].
Corollary 1.4. Let S(x1 , x2 ; k; n) denote the number of overpartitions of n into parts not divisible by 3 with xj parts congruent to j modulo 3 and k non-overlined parts. Let T (x1 , x2 ; k; n)denote the number of overpartitions of n with xj parts congruent to j or 3 modulo 3 and k
non-overlined parts, where parts differ by at least 3 if the smaller is overlined OR both parts
are divisible by 3, and parts differ by at least 6 if the smaller is overlined AND both parts aredivisible by 3. Then S(x1 , x2 ; k; n) = T (x1 , x2 ; k; n).
Corollary 1.5. Let G(n) be the number of overpartitions of n into parts congruent to 2, 4, or
5 modulo 6. Let H(n) be the number of overpartitions of n such that (i) there are no ones,(ii) 3 or 3 may occur if 2 does, (iii) the difference between consecutive parts is at least 6 if thesmaller is overlined OR both parts are congruent to 0 modulo 6, both are 1 modulo 6, or both are
3 modulo 6, (iv) the difference between consecutive parts is at least 12 if the smaller is overlined
AND both are 0 modulo 6, both are 1 modulo 6, or both are 3 modulo 6, and (v) any maximal
set of consecutive parts, {λi , λi+1 , . . . , λi+k }, where each part is congruent to 5 modulo 6 and thedifference between any two consecutive parts is the minimum allowed by conditions (iii) − (iv),must have even length. Then G(n) = H(n).
For another application we present some weighted overpartition theorems, along the lines of[1, 2]. Here we define a chain in an overpartition to be a maximal set of consecutive parts
where the difference between consecutive parts is 0 if the smaller is non-overlined and 1 if it
is overlined. Then any overpartition may be uniquely decomposed into chains. For example,
the overpartition (5, 5, 5, 4, 3, 2, 1) is comprised of the chains (5, 5, 5, 4), (3, 2), and (1). In the

A THEOREM ON SEVEN-COLORED OVERPARTITIONS AND ITS APPLICATIONS

3

theorems below, we will define a weight on chains, the weight of the overpartition being the
product of the weights of its chains. Denote by κ the length of a chain and σ the smallest partin the chain.
Corollary 1.6. Starting with k = 0, let a(k) be the 6-periodic sequence that begins 1, 1, 0, −1, −1, 0, ...
Then the number of overpartitions of n into parts not divisible by 3 is equal to the number ofweighted overparititons of n where the weight of a chain is
(
a(κ),
σ=1
a(κ) + a(κ − 1), σ ≥ 2Corollary 1.7. The number of overpartitions of n into parts divisible by 3 is equal to the numberof overpartitions of n where the weight of a chain is−1, κ ≡ 1 (mod 3) and σ ≥ 20,κ ≡ 1 (mod 3) and σ = 11,otherwise
As a final application, we relate a weighted count of the overpartitions of n to the number ofrepresentations of n as the sum of three squares. For an overpartition λ, define f (λ) to be 0,
if 1 and 1 do not occur, and the length of the chain containing 1 and/or 1, otherwise. We also
define r3 (n) to be the number of integer solutions to x2 + y 2 + z 2 = n.
Corollary 1.8. The number of overpartitions λ of n, each weighted by (−1)f (λ) (2f (λ) + 1), isequal to (−1)n r3 (n).
We should point out that r3 (n) is also related to class numbers of imaginary quadratic fields[14, p.389-390].
2. Proof of Lemma 1.1
Although the identity (1.3) may appear daunting, it is happily only a slight extension of thework in [6, Section 4]. Indeed, the case d = 0 of Lemma 1.1 is Eq. 1.4 of [6]. For the proof of(1.3), we shall require the 6 ψ6 summation [12, p.239, Eq. (II.33)],
X (1 − aq 2n )(b, c, d, e)n (qa2 /bcde)n(aq, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de, q, q/a)∞=,(1 − a)(aq/b, aq/c, aq/d, aq/e)n(aq/b, aq/c, aq/d, aq/e, q/b, q/c, q/d, q/e, qa2 /bcde)∞
n∈Z

(2.1)
and a certain 8 φ7 transformation [12, p.243, Eq. (III.24)],
X (1 − aq 2n )(a, b, c, d, e, f )n (a2 q 2 /bcdef )n(aq, b, bcµ/a, bdµ/a, beµ/a, bf µ/a)∞=(1 − a)(q, aq/b, aq/c, aq/d, aq/e, aq/f )n(aq/c, aq/d, aq/e, aq/f, µq, bµ/a)∞n≥0×X (1 − µq 2n )(µ, aq/bc, aq/bd, aq/be, aq/bf, bµ/a)n bn
n≥0(1 − µ)(q, bcµ/a, bdµ/a, beµ/a, bf µ/a, aq/b)n

.

(2.2)

Here we require µ = a3 q 2 /b2 def . Also, we have extended (1.2) to all integers with the relation(a)−n =(−1)n q n(n+1)/2.
an (q/a)n

(2.3)

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JEREMY LOVEJOY

We begin with a constant term lemma, employing the usual notationX[z 0 ]
a(n)z n = a(0).

(2.4)

Lemma 2.1.
[z 0 ](−zq, −1/z, −abq/z, −acq/z, −bcq/z, q, adq, bdq, cdq)∞ (1 + abc/z 2 )= 1.
(a/z, b/z, c/z, dzq, −abcdq/z, −aq, −bq, −cq, −dq)∞(2.5)

Proof. We begin by using the 6 ψ6 summation as above, with a = −1/z, b = −1/a, c = −1/b, d =−1/c, and e = 1/dz. Then the constant term identity becomes
1 = [z 0 ]X (1 + q 2n /z)(−1/a, −1/b, −1/c, 1/dz)n (−abcdq/z)n(1 + abc/z 2 )(1 − a/z)(1 − b/z)(1 − c/z)
(aq/z, bq/z, cq/z, −dq)nn∈ZX (1 + q 2n /z)(−1/a, −1/b, −1/c, 1/dz)n (−abcdq/z)n(1 + abc/z 2 )= [z 0 ](1 − a/z)(1 − b/z)(1 − c/z)
(aq/z, bq/z, cq/z, −dq)nn≥0+ [z 0 ](1 + abc/z 2 )(1 − a/z)(1 − b/z)(1 − c/z)

X (1 + q 2n /z)(−1/a, −1/b, −1/c, 1/dz)n (−abcdq/z)nn≤−1(aq/z, bq/z, cq/z, −dq)n.

Now the constant term in the middle line is obviously 1. Replacing n by −n, the last line canbe written as∞

X (1 + zq 2n )(zq/a, zq/b, zq/c, −q/d)n−1 (−abcdq/z)n−(1 + z 2 /abc)(1 + 1/d)
(1 + aq)(1 + bq)(1 + cq)(1 − zdq)(−aq 2 , −bq 2 , −cq 2 , zdq 2 )n−1=

(z + abc/z)(dq)(1 + 1/d)
(1 + aq)(1 + bq)(1 + cq)(1 − zdq)

n=1
∞
X
n=0

(1 + zq 2n+2 )(zq/a, zq/b, zq/c, −q/d)n (−abcdq/z)n.
(−aq 2 , −bq 2 , −cq 2 , zdq 2 )nIn the 8 φ7 transformation (2.2) we make the substitutions a = −zq 2 , b = q, c = zq/a, d =zq/b, e = zq/c, and f = −q/d. Then µ = abcdq 2 , and the last expression above becomes(−zq 2 , q, −bcdq 2 , −acdq 2 , −abdq 2 , abcq 2 /z)∞(−aq, −bq, −cq, dzq, abcdq 2 , −abcdq/z)∞
X (1 − abcdq 2n+2 )(abcdq 2 , −aq, −bq, −cq, dzq, −abcdq/z)n q n.
×(q, −bcdq 2 , −acdq 2 , −abdq 2 , abcq 2 /z, −zq 2 )n(dq + q)(z + abc/z) ×n≥0

Observe that this entire expression is negated under the substitution z → −abc/z, which showsthat its constant term is 0 and completes the proof of the lemma.
Proof of Lemma 1.1. Now to prove Lemma 1.1, we change it to a constant term statement:(−aq, −bq, −cq)∞=(2.6)(adq, bdq, cdq)∞X
q s+Tδ +T +Tφ−1 +` (1 − q α (1 − q φ ))aα+δ++` bβ+δ+φ+` cγ++φ+` ds+` (−1/d)s (−1)`[z 0 ].
z α+β+γ+δ++φ+`−s (q)α (q)β (q)γ (q)δ (q) (q)φ (q)`s∈Z
α,β,γ,δ,,φ,`≥0

A THEOREM ON SEVEN-COLORED OVERPARTITIONS AND ITS APPLICATIONS

5Each of the eight sums on the right can be written as an infinite product by using special
cases of [12, p.239, Eq. (II.29)]; the sum over s byX(−zq, −1/z, q)∞(−1/a)n (azq)n =,(azq, −aq)∞n∈Z

the sums over α, β, γ, and ` by
X zn
n≥0

(q)n

=

1
,
(z)∞

and the sums over δ, , and φ by
X z n q Tn
n≥0

(q)n

= (−zq)∞ .

After a bit of simplification, this gives[z 0 ](−zq, −1/z, −abq/z, −acq/z, −bcq/z, q, adq, bdq, cdq)∞ (1 + abc/z 2 )= 1,(a/z, b/z, c/z, dzq, −abcdq/z, −aq, −bq, −cq, −dq)∞

which is precisely Lemma 2.1.

(2.7)


3. Proof of Theorem 1.3
First we recall from [6] that the seriesX
q Ts +Tδ +T +Tφ−1 (1 − q α (1 − q φ ))aα+δ+ bβ+δ+φ cγ++φ
α,β,γ,δ,,φ≥0
s=α+β+γ+δ++φ(q)α (q)β (q)γ (q)δ (q) (q)φ(3.1)

is the generating function for those partitions λ counted by C(α, β, γ, δ, , φ, 0; 0; n). For sucha λ, we remove a “staircase”, i.e., 1 from the smallest part, 2 from the next smallest, and soon, to get a partition λ0 . In terms of generating functions, this corresponds to removing theterm q Ts from (3.1). We then take an ordinary partition µ into positive parts having color abcd
and weighted by −1 if the number of parts is odd, and put its parts into λ0 according to theorder (1.4). This corresponds to multiplying (3.1) by (−1)` q ` /(q)` . Finally we turn λ0 into an
overpartition by adding on a “generalized” staircase corresponding to the generating function(dq)s (−1/d)s , where s is now α + β + γ + δ +  + φ + `. This is done by augmenting each partby one and then adding, for each j occurring in the partition ν generated by (−1/d)s , one toeach of the first j parts and overlining the j + 1st part. The result is an overpartition countedby C(α, β, γ, δ, , φ, `; k; n), and the generating function is the left hand side of (1.3). By theidentity (1.3), the theorem is proved.

4. Proofs of the Corollaries
Proof of Corollary 1.4. Set c = 0, q = q 3 , a = aq −2 , and b = bq −1 in Theorem 1.3. On the
product side, we haveX(−aq, bq 2 ; q 3 )∞=S(x1 , x2 ; k; n)ax1 bx2 dk q n .
(adq, bdq 2 ; q 3 )∞x1 ,x2 ,k,n≥0

6

JEREMY LOVEJOY

On the sum side, when c = 0, we are left with only 3 colors, ab, a, and b. Under the remaining
substitutions, na becomes 3n − 2, nb becomes 3n − 1, and nab becomes 3n − 3. It is easy tocheck that the difference conditions (iii) and (iv) in Definition 1.2 translate into those defining
T (x1 , x2 ; k; n). Finally, since there are no abcd-parts, the exponent of d is just the number of
non-overlined parts.
Proof of Corollary 1.5. Here we make the substitutions q = q 6 , d = 1, a = q −4 , b = q −2 , andc = q −1 in Theorem 1.3. On the product side, we have(−q 2 , q 4 , q 5 ; q 6 )∞ X=
G(n)q n .
(q 2 , q 4 , q 5 ; q 6 )∞n≥0

On the sum side, na becomes 6n − 4, nb becomes 6n − 2, nc becomes 6n − 1, nab becomes 6n − 6,nac becomes 6n − 5, nbc becomes 6n − 3, and nabcd becomes 6n − 7. Conditions (i) and (ii) inthe definition of H(n) correspond to conditions (i) and (ii) in Definition 1.2.
For the difference conditions, let us first focus on the parts congruent to 5 modulo 6. One
can check that the required minimum difference between a part not congruent to 5 modulo 6
and one that is congruent to 5 modulo 6 is the same whether the part congruent to 5 modulo 6
came from an abcd part or a c part. In the case of two parts congruent to 5 modulo 6, we havefour possibilities, since each came from either a c part or an abcd part. Again, it is easy to checkthat the required minimum differences are equal, except when the larger came from a c part andthe smaller from an abcd part, in which case the required minimum difference is 6 more thanin the other cases. Hence, a any maximal set of consecutive parts, {λi , λi+1 , . . . , λi+k }, where
each part is congruent to 5 modulo 6 and the difference between any two consecutive parts isthe minimum allowed by conditions (iii) − (iv) defining H(n), must consist of a string of abcd
parts followed by a string of c parts. Given the weight of −1 to the number of abcd parts, this
implies that such a maximal sequence must have even length.
Finally, it is one more time easily verified that the minimum required differences in conditions(iii) and (iv) in the definition of H(n) follow from the conditions (iii) and (iv) in Definition
1.2.

Proof of Corollary 1.6. Set c = 0, d = 1, a = −e2πi/3 , and b = −e4πi/3 in Theorem 1.3. The
infinite product becomes(−q)∞ (q 3 ; q 3 )∞,(q)∞ (−q 3 ; q 3 )∞the generating function for overpartitions into parts not divisible by 3. On the sum side, only
the colors a, b, and ab remain. A chain, from largest to smallest, must consist of a string of bparts followed by a string of a parts, possibly followed by an ab part. Suppose for a momentthat there is no ab part in a chain of length k. Then, the weight of the chain will bekX

aj bk−j= [xk ]

j=0

= [xk ]

1
(1 − ax)(1 − bx)
1(1 + e2πi/3 x)(1 + e4πi/3 x)= [xk ](1 + x − x3 − x4 + x6 + x7 − x9 − x10 + x11 · · · )= a(k).

A THEOREM ON SEVEN-COLORED OVERPARTITIONS AND ITS APPLICATIONS

7

So, taking into account the possibility of an ab part in the chain, the weight of a chain will bea(k) + a(k − 1), unless the smallest part of the chain is one, in which case there can be no abpart and then weight is just a(k).
Proof of Corollary 1.7. Set b = d = 1, a = −e2πi/6 , and c = −e−2πi/6 in Theorem 1.3. On
the product side we get (−q 3 ; q 3 )∞ /(q 3 ; q 3 )∞ , the generating function for overpartitions whoseparts are divisible by 3. On the sum side, we want to determine the weight of a chain. If a chain
has no ab parts, no ac parts, and no bc parts, then from largest to smallest, it must consist ofa string of c parts followed by a string of b parts followed by a string of a parts followed by astring of abcd parts. The weight of such a chain is[xk ]1(1 + e2πi/6 x)(1 + e−2πi/6 x)(1 + x)(1 − x)

= [xk ]

1
(1 − x3 )(1 + x)

= [xk ](1 − x + x2 + x6 − x7 + x8 + x12 · · · ).
Call the coefficient of xk in the final expression b(k). Now suppose that the smallest part ofthe chain is at least 2 and that the chain does have one or more parts of binary color. By theordering (1.4), the chain could end with one part in binary color in 3 ways, ab, ac, or ab; it could
end in two parts with binary color in 3 ways, (bc, ab), (ac, ab), or (bc, ac); or it could end withthree parts of binary color in one way, (bc, ac, ab). Hence the weight of a chain whose smallest
part is at least two is b(k) + b(k − 3).
Next suppose that the smallest part of a chain is equal to 1. Now ab, bc, ac, and abcd may
not be the color of the smallest part, unless 1a occurs in which case the smallest part may
be 1bc . Because abcd is not the smallest part in the chain, there are no abcd parts in the
chain. HenceP
if 1bc is not the smallest part, the weight of the chain is the coefficient of xk in
3
1/(1 − x ) = c(k)q k . To allow for the possibility of a 1bc preceding a 1a , the weight of such a
chain is c(k − 2) + c(k).
We conclude with the proof of Corollary 1.8, which itself is a corollary of the following:
Corollary 4.1. For ` ∈ Z, the coefficient of z ` q n in(q, −zq, −q/z)∞(−q, zq, q/z)∞(4.1)

is equal to the number of overpartitions λ of n having f (λ) ≥ |`|, each weighted by (−1)`+f (λ) .
Proof. Let a = −1, b = z, c = z, and d = 1 in Theorem 1.3. The product is (4.1), and weinterpret the sum side as in the previous two corollaries using weights of chains. If a chain oflength k has no ab, ac, or bc parts, then its weight is the coefficient of xk in
1.
(1 + x)(1 − zx)(1 − x/z)(1 − x)If the smallest part of the chain is at least 2, then, allowing for the possibility of ab, ac, and bc
parts, the weight is the coefficient of xk in
1 + x(1 − z − 1/z) + x2 (1 − z − 1/z) + x3
1=,
(1 + x)(1 − zx)(1 − x/z)(1 − x)
1−x8

JEREMY LOVEJOY

which is 1. Hence the overall weight of an overpartition depends only on the chain whose smallest
part is 1 (or 1). As before, the weight of such a chain is the coefficient of xk in(1 − x),(1 − xz)(1 − x/z)
which iskX(−1)`+k z ` .

`=−kNotice that if we set z = 1 in the above corollary, then the product becomes the generating
function for overpartitions. If we set z = −1, then the product becomes (q)3∞ /(−q)3∞ , and we
can deduce Corollary 1.8:
Proof of Corollary 1.8. Let z = −1 in Corollary 4.1. Then each overpartition λ is weighted
byX(−1)f (λ)
1 = (−1)f (λ) (2f (λ) + 1).
|`|<f (λ)

Since

∞
X
(q)∞
2
=
(−1)n q n ,(−q)∞ n=−∞the corollary follows.


5. Concluding Remarks

To finish we would like to offer some suggestions for future research. First, exhibit further
overpartition (or partition) theorems that follow from Theorem 1.3. Second, try to find “finite”
versions of Lemma 1.1 along the lines of [3]. Third, examine some of the other proofs of Schur’stheorem and see if they can be extended to overpartitions. Fourth, find a combinatorial proofof Theorem 1.3 (or any of its corollaries). Finally, study the 11-color partition theorem in [4]and see if it has some kind of overpartition analogue.
References[1] K. Alladi, Partition identities involving gaps and weights, Trans. Amer. Math. Soc. 349 (1997), 5001-5019.
[2] K. Alladi, Partition identities involving gaps and weights II, Ramanujan J. 2 (1998), 21-37.
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[8] G.E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 49 (1984), no. 301.
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A THEOREM ON SEVEN-COLORED OVERPARTITIONS AND ITS APPLICATIONS

9[11] S. Corteel, J. Lovejoy, and A. Yee, Overpartitions and generating functions for generalized Frobenius partitions, in Mathematics and Computer Science III. Algorithms, Trees, Combinatorics, and Probabilities, 15-24,
Birkhauser, 2004.
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[14] W. Kohnen and K. Ono, Indivisibility of class numbers of imaginary quadratic fields and orders of TateShafarevich groups of elliptic curves with complex multiplication, Invent. Math. 135 (1999), 387-398.
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[16] J. Lovejoy, Overpartition theorems of the Rogers-Ramanujan type, J. London Math. Soc. 69 (2004), 562-574.
[17] I. Schur, Zur additiven Zahlentheorie, Gessamelte Abhandlungen 2, Springer, Berlin (1973), 43-50.
CNRS, LIAFA, Université Denis Diderot, 2, Place Jussieu, Case 7014, F-75251 Paris Cedex 05,
FRANCE
E-mail address: lovejoy@liafa.jussieu.fr