n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND
ROGERS-RAMANUJAN IDENTITIESJEREMY LOVEJOY AND OLIVIER MALLET
Abstract. In the early 90’s Andrews discussed a certain q-series whose coefficients are determined by a twisted divisor function. We provide several other examples of this nature. All ofthese q-series can be interpreted combinatorially in terms of n-color overpartitions, as can someclosely related series occurring in identities of the Rogers-Ramanujan type.

1. Introduction
In 1988 Andrews, Dyson, and Hickerson [16] made an extensive study of a q-series that firstcame to light in Ramanujan’s lost notebook [12, 13],Xn≥0

q n(n+1)/2= 1 + q − q 2 + 2q 3 − 2q 4 + q 5 + q 7 − 2q 8 + 2q 10 + · · · .
(1 + q)(1 + q 2 ) · · · (1 + q n )(1.1)

If r(n) denotes the coefficient of q n in this series, then r(n) has a rather simple combinatorialinterpretation - as the number of partitions of n into distinct parts with even rank minus thenumber with odd rank. Recall that the rank of a partition is the largest part minus the numberof parts. On the other hand, Andrews, Dyson, and Hickerson showed that r(n) is almost always
0 and assumes every integer infinitely often, facts which may√ be deduced from a multiplicative“exact” formula relating r(24n + 1) to the arithmetic of Z[ 6].
Subsequently, in a study again related to Ramanujan’s lost notebook, Andrews [14, 15] wroteof another series which was “quite reminiscent” of (1.1), having multiplicative coefficients which
are almost always 0. This series is
X q n2 (1 − q)(1 − q 2 ) · · · (1 − q n−1 )n≥1(1 − q n+1 )(1 − q n+2 ) · · · (1 − q 2n )= q + q 3 + q 4 + 2q 7 + q 9 + q 12 + 2q 13 + · · · .

(1.2)

The “mystery” of this series was immediately explained by Andrews himself, who showed thatthe multiplicativity is dictated by a twisted divisor function through the identity
X X µd¶X (q)n−1 q n2

 qn.
=
(1.3)
(q; q 2 )n (−q)n
3
n≥1

n≥1

d|n

Date: July 25, 2007.
2000 Mathematics Subject Classification. 11P81, 05A17, 33D15.
Key words and phrases. partitions, overpartitions, n-color overpartitions, real quadratic fields, Bailey pairs,
Rogers-Ramanujan identities.
The first author was partially supported by an ACI “Jeunes Chercheurs et Jeunes Chercheuses”.
1

2

JEREMY LOVEJOY AND OLIVIER MALLET

Here we have employed the usual Legendre symbol and rewritten Andrews’ series using thestandard q-series notation [23].
The first goal of this paper is to exhibit several other q-series which resemble (1.2) bothin their q-series definitions and in their having multiplicative coefficients arising from twisted
divisor functions.
Theorem 1.1. We have
X (q)n−1 (−1)n q n(n+1)/2(q; q 2 )n (−q)n

n≥1

=−

X
n≥1



X µd¶

 qn,
5

(1.4)

d|n


X (q)n (−1)n q n(n+1)/2=(q; q 2 )n+1 (−q)n

n≥0

and
X
n≥0

X
n≥0


X µd¶

 qn,
5
d|5n+1



2
X
X µd¶
(q)n q n +n

 qn.
=
(q; q 2 )n+1 (−q)n
3
n≥1

(1.5)

(1.6)

d|3n+1The proofs of these identities are presented in the next section using Bailey pairs. Equation(1.6) is rather straightforward, while for the remaining two we shall use Bailey pairs related toseventh order mock theta functions and pass through real quadratic fields, as in [16, 20, 25].
Of course identities like these are most fully appreciated when their combinatorial implications
are well understood. Thus the second goal of this paper is to develop the combinatorics necessary
to interpret the q-series occurring in (1.3) - (1.6) in terms of n-color overpartitions. These extend
the n-color partitions whose study was initiated in the 80’s by Agarwal, Andrews, and Bressoud[1, 3, 4, 7, 8, 9], inspired by work of Baxter [7, Section 3]. The basic definitions and fundamental
generating functions (Theorems 3.1 and 3.2) for n-color overpartitions are contained in Section
3.
In Section 4 we then present the combinatorial versions of identities (1.3) - (1.6) as Corollaries
4.5 - 4.8. We shall also be able to give nice combinatorial interpretations of some identities of
the Rogers-Ramanujan type, to which the series in (1.3) - (1.6) are closely related. These are
X (−1)n q n2(−q)∞ (q 3 ; q 3 )∞=,(q; q 2 )n (q)n(q)∞ (−q 3 ; q 3 )∞

(1.7)

n≥0

X (−1)n q n(n+1)/2
n≥0(q; q 2 )n (q)nX (−q)n q n(n+1)/2
n≥0

and

(q; q 2 )n+1 (q)n=(−q)∞ (q 5 ; q 5 )∞,(q)∞ (−q 5 ; q 5 )∞(1.8)(−q)∞ (q 3 , q 7 , q 10 ; q 10 )∞
,
(q)∞

(1.9)

=

X (−q)n q n2 +n(−q)∞ (q, q 5 , q 6 ; q 6 )∞=,(q; q 2 )n+1 (q)n
(q)∞

n≥0

(1.10)

n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND ROGERS-RAMANUJAN IDENTITIES 3

which are [14, Eq. (4.6)],[29, Eq. (5)], [30, Eq. (45)], and [30, Eq. (22)], respectively. The
combinatorial versions of these identities are Corollaries 4.1 - 4.4. We close the paper with some
remarks and ideas for future projects.
2. Proof of Theorem 1.1
Here we prove the three identities in Theorem 1.1 using the method of Bailey pairs. Recall
that two sequences (αn , βn ) are said to form a Bailey pair with respect to a if for all n ≥ 0 we
have
n
X
αn
.
(2.1)
βn =
(q)n−r (aq)n+r
r=0

The following allows us to prove identities using Bailey pairs:
Lemma 2.1 ([17], Cor. 2.1). If (αn , βn ) is a Bailey pair with respect to a, then∞X∞(ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n βn =n=0(aq/ρ1 )∞ (aq/ρ2 )∞ X (ρ1 )n (ρ2 )n (aq/ρ1 ρ2 )n αn,(aq)∞ (aq/ρ1 ρ2 )∞(aq/ρ1 )n (aq/ρ2 )n

(2.2)

n=0

provided both sides converge absolutely.
We begin by treating the last equation in Theorem 1.1, which is the easiest. By [15, Eq.
(3.16),(3.17)], the sequences (αn , βn ) form a Bailey pair with respect to q, where
αn =(−1)n−1 (2n + 1)q n(n+1)/2(1 − q)andβn = −(q)n.
2(q; q )n+1 (−q)nInserting this into Lemma 2.1 with a = q and ρ1 , ρ2 → ∞ yields
2X
1 X(q)n q n +n=(−1)n (2n + 1)q 3n(n+1)/2(q; q 2 )n+1 (−q)n
(q)∞
n≥0

=

n≥0
3
3
(q ; q )3∞

(q)∞

,

the last equality following from a well-known identity of Jacobi. Now, the modular formq(q 9 ; q 9 )3∞ /(q 3 ; q 3 )∞ is the twisted Eisenstein series whose coefficient of q 3n+1 is precisely thecoefficient of q n on the right hand side of (1.6). So this identity is established.
¤
We turn to (1.5), which is somewhat more involved. We begin with a Bailey pair with respect
to q related to the seventh order mock theta functions [11, Lemma 12, i = 2],


n−1
X
1  3n2 +n X −j 222qq
+ 2q 3n +2nq −j −j  ,
α2n =
1−q
j=0

|j|≤n


α2n+1 = −1  3n2 +4n+1
2q
1−q

n
X
j=0


2

2

q −j −j + q 3n +5n+2

X
|j|≤n

2

q −j  ,4

JEREMY LOVEJOY AND OLIVIER MALLET

and
βn =

1
(q; q 2 )n+1 (−q)n.

Inserting this Bailey pair into Lemma 2.1 with ρ1 = q and ρ2 → ∞ and simplifying gives
X (q)n (−1)n q n(n+1)/2
n≥0(q; q 2 )n+1 (−q)n

=

X

2

2

q 5n +2n−j +

n≥0
|j|≤n

+

X

2

q 5n +8n+3−j

2

n≥0
|j|≤n

X n−1
X

2

2

q 5n +3n−j −j +n≥0 j=−n

n
X X

(2.3)
2

2

q 5n +7n+2−j −j .

n≥0 j=−n−1

We shall now proceed to demonstrate that the right hand side above is precisely the generating
functionfor the number of inequivalent elements of√√ (positive) norm 5k+1 in the ring of integers√ ofQ( 5). Recall that this ring of integers is Z[(1+ 5)/2], and the norm function is N (x+y 5) =x2 − 5y 2 . The solutions to x2 − 5y 2 = 5k + 1 come in two flavors: x and y integers with x ≡ ±1
(mod 5) or x and y half integers with 2y odd and 2x ≡ ±3 (mod 10). It turns out that the firsttwo sums above correspond to solutions of the first type and the second two sums to solutionsof the second type.
√√To see this, first note that√the element x + y 5 of Z[(1 + 5)/2] whose norm is 1 with x andy minimal positive is 3/2 + 5/2. Hence, by [16, Lemma√3], for any m > 0 each equivalenceclass of solutions of u2 − 5v 2 = m contains a unique u + v 5 withu > 0 and − u < v ≤ u.

(2.4)

Next, in the first two sums on the right hand side of (2.3), we let q = q 5 and multiply by q.
The exponent of q in the first sum is (5n + 1)2 − 5j 2 and in the second sum it is (5n + 4)2 − 5j 2 .
As mentioned above, these are the two possible ways to represent a number of the form 5k + 1
as x2 − 5y 2 with x and y integers. Moreover, replacing u by 5n + 1 or 5n + 4 and v by j in (2.4),
we see that the inequalities n ≥ 0 and |j| ≤ n in the two sums guarantee that we have at mostone element of norm 5k + 1.
The treatment of the second two sums is similar, so we omit the details. Here one regards
the solution to x2 − 5y 2 = 5k + 1 with x and y half-integers as a solution to x2 − 5y 2 = 20k + 4
with y odd and x ≡ ±3 (mod 10). Then q is replaced by q 20 , a factor of q 4 is multiplied, andone proceeds as above.
To finish the proof of identity√ (1.5), we may quote from [18] that if a(m) denotes the number of
elements of norm m in Z[(1+ 5)/2], then (i) a(mn) = a(m)a(n) if m and n are relatively prime,(ii) a(5r ) = 1, (iii) a(pr ) = r + 1 for primes p ≡ ±1 (mod 5) and (iv) a(pr ) = (1 + (−1)r )/2
for primes p ≡ ±2 (mod 5). It is easy
¡ ¢ to see that this is equivalent to counting divisors d of m¤weighted by the Legendre symbol d5 .
Finally, we turn to the first identity in Theorem 1.1. We begin again with a Bailey pair related
to the seventh order mock theta functions, this time a Bailey pair with respect to 1 [11, Lemma
12, i = 0],
X
X
2
2
2
2
α2n = q 3n +n
q −j − q 3n −nq −j ,
|j|≤n

|j|<n

n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND ROGERS-RAMANUJAN IDENTITIES 5

α2n+1 = −2q

3n2 +4n+1

nXq−j 2 −j+q

3n2 +2n

j=0

n−1
X

2

q −j −j ,

j=0

and

1
.
(q; q 2 )n (−q)nInstead of using this pair directly in Lemma 2.1, we first transform it using the following lemma:
βn =Lemma 2.2 (Lemma 2.2, [25]). If (αn , βn ) is a Bailey pair with respect to 1, then (αn∗ , βn∗ ) is a
Bailey pair with respect to q, where α0∗ = β0∗ = 0, and for n ≥ 1,
βn∗ =
and
(−1)n q n(n−1)/2 (1 − q 2n+1 )αn∗ =(1 − q)

Ã
n+

−1
βn
(1 − q n )

(2.5)

n
X
(−1)r+1 q −r(r−1)/2 αr
r=1(1 − q r )

+2

n
X
r=1

qr
(1 − q r )

!
.

(2.6)

Now if we insert our Bailey pair into Lemma 2.2 and then put this new pair into Lemma
2.1 with ρ1 = q and ρ2 → ∞, then on the “β side”, we have the negative of the left hand sideof (1.4). For the “α side”, leaving the αn∗ in the form (2.6) for now, notice that we obtain an
expression of the form
n
X 2
X
n
2n+1
q (1 − q
)
fr ,
r=1

n≥1

where the fr comes from the parentheses in (2.6). Thus this collapses to
fn = 1 +Pn≥1 qn2 f , withn(−1)n+1 q −n(n−1)/2 αn2q n+.
(1 − q n )(1 − q n )Now we insert the αn we started with and after some simplification we may obtain the identity−

X (q)n−1 (−1)n q n(n+1)/2n≥1(q; q 2 )n (−q)n

=

n
X X

2

2

q 5n −j +n≥1 j=−n+1

XX

2

2

q 5n +5n+1−j −j .

(2.7)

n≥0 |j|≤n

Now we proceed as in the proof of (1.5) above. We shall discover that the right hand side of(2.7) is theelements of negative norm −m√ generating function for the number of (inequivalent)√in Z[(1+ 5)/2]. Since this ring contains the element 1/2+ 5/2 of norm −1, this is the same ascounting elements of positive norm m. Then citing the formula of [18] as in the last paragraphof the proof of (1.5), we will have established (1.4).
So, we again quote Lemma 3 of [16], which tells us that for√each equivalence class of elementsof negative norm −m, there is a unique representative j + n 5 withn > 0 and − n < j ≤ n.

(2.8)

But these are precisely the inequalities in the first sum of (2.7). This then takes care of theelements of norm −m where this unique representative has j and n integral. The second sum
takes care of the case when j and n are half-integers. To see this, we rewrite the exponent of qas 5((2n + 1)/2)2 − ((2j + 1)/2)2 . Then replacing n by (2n + 1)/2 and j by (2j + 1)/2 in (2.8)

6

JEREMY LOVEJOY AND OLIVIER MALLET

gives n ≥ 0 and |j| ≤ n, which are precisely the inequalities in our second sum. This completes
the proof of (1.4) and of Theorem 1.1.
3. n-color overpartitions
In this section we define the basic notions associated with n-color overpartitions and determinesome basic generating functions. An n-color partition is a partition in which each number n
may appear in n colors, with parts ordered first according to size and then according to color.
For example, there are 13 n-color partitions of 4,(44 ), (43 ), (42 ), (41 ), (33 , 11 ), (32 , 11 ), (31 , 11 ), (22 , 22 ), (22 , 21 ), (21 , 21 ),(22 , 11 , 11 ), (21 , 11 , 11 ), (11 , 11 , 11 , 11 ).
Motivated by Baxter’s solution of the hard hexagon model (see [7, Section 3]), the study ofn-colored partitions was initiated in the 1980’s by Agarwal, Andrews, and Bressoud. These
partitions arose in identities of the Rogers-Ramanujan type (e.g. [1, 3, 4, 7]) and they played an
influential role in the development of the Bailey lattice [8, Section 5]. They have been related toa number of other combinatorial objects, such as lattice paths, generalized Frobenius partitions,and plane partitions (e.g. [2, 5, 6, 9]). Indeed, the latter have the same generating function as
unrestricted n-color partitions,Y1= 1 + q + 3q 2 + 6q 3 + 13q 4 + 24q 5 + · · ·(1 − q n )nn≥1

An exposition of much of the above, as well as further results on n-color partitions, may befound in the book Partition Theory [10].
We define an n-color overpartition to be an n-color partition in which the final occurrence ofa part nj may be overlined. For example, there are 16 n-color overpartitions of 3,(33 ), (32 ), (31 ), (33 ), (32 ), (31 ), (22 , 11 ), (21 , 11 ), (22 , 11 ), (22 , 11 ),(22 , 11 ), (21 , 11 ), (21 , 11 ), (21 , 11 ), (11 , 11 , 11 ), (11 , 11 , 11 ).
The generating function for n-color overpartitions is clearly
Y (1 + q n )n
= 1 + 2q + 6q 2 + 16q 3 + 38q 4 + 88q 5 + · · ·(1 − q n )n

(3.1)

(3.2)

n≥1

We shall be interested in n-color overpartitions whose weighted difference is bounded below.
The weighted difference of two adjacent parts mi and nj , denoted ((mi , nj )), is defined to bem − i − n − j. For an n-colored overpartition λ = (λ1 , λ2 , . . . , λs ), we write c(λj ) for the color of
λj . The next theorem provides the n-color overpartition-theoretic basis for interpreting q-series
which contain the term 1/(q; q 2 )n .
Theorem 3.1. If we define fm (r, s, k) byX
r,s,k≥0fm (r, s, k)ar bs q k :=(−a)m q m(m+1)/2,(q; q 2 )m (bq)m(3.3)

then fm (r, s, k) is the number of n-color overpartitions λ of k into m parts such that (i) the
weighted differences satisfy ((mi , nj )) ≥ −1 + χ(nj overlined), (ii) r is the number of overlined

n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND ROGERS-RAMANUJAN IDENTITIES 7

parts, and(iii)s = λ1 − c(λ1 ) +X(χ(λj non-overlined) − 2c(λj )) .

j≥2

Proof. The proof is by classical methods of constructive partition theory. On the right sideof (3.3), the term q m(m+1)/2 /(bq)m is the generating function for partitions µ1 into m distinct
parts, where the exponent of b tracks the largest part minus m. Notice that if we assign thecolor 1 to each part, then µ1 is an n-color (over)partition whose weighted differences are at least−1, with equality if an only if two adjacent parts are consecutive integers.
Now the factor 1/(q; q 2 )m contributes a partition µ2 into odd parts at most 2m − 1. These
are added to µ1 in the following way: For a part 2t + 1 of µ2 , we add 2 to each of the first t
largest parts of µ1 and then to the t + 1st largest part we add 1 to both the part size and thecolor. It is clear that this operation leaves all weighted differences fixed. It is also easy to seethat this procedure is invertible, since the colors indicate exactly where an odd part has been
added.
Finally, the factor (−a)m gives a partition µ3 into distinct non-negative parts less than m,where the exponent of a counts the number of parts. We add this bijectively to our object in
the usual way, for each part t of µ3 adding 1 to the t largest parts and then overlining the next
part. This doesn’t change the weighted differences except when the smaller part picks up an
overline. This is accounted for by the characteristic function in condition (i).
To finish the proof, it is clear that r is the number of overlined parts and the number of parts
is m. In order to see why condition (iii) holds, we defineX
σ = λ1 − c(λ1 ) +(χ(λj non-overlined) − 2c(λj )) .
j≥2

In µ1 , we have s = λ1 − m = σ since there are no overlined parts and all the colors are equal
to one. When we apply µ2 , every part 2t + 1 adds 2 to λ1 and 1 to both λt+1 and its color. If
t = 0, both λ1 and c(λ1 ) increase by 1, so σ does not change; if t > 0, λ1 increases by 2 andc(λt+1 ) increases by 1, so σ does not change either. When we apply µ3 , every part t > 0 adds 1
to λ1 and overlines λt+1 , so σ is not modified. If t = 0, λ1 is not modified; it becomes overlined,
but since the summation in the definition of σ begins at j = 2, this does not change the valueof σ. Hence, we have s = σ in the final n-color overpartition, so condition (iii) is verified.
¤
For series with the term 1/(q; q 2 )n+1 , we shall consider (n + 1)-color overpartitions. Here thenumber n may occur in any of the n + 1 colors {1, 2, . . . , n + 1}.
Theorem 3.2. If we define gm (r, s, k) by
X(−aq)m q m(m+1)/2
gm (r, s, k)ar bs q k :=,(q; q 2 )m+1 (bq)m(3.4)

r,s,k≥0then gm (r, s, k) is the number of (n + 1)-color overpartitions λ of k into m + 1 non-negative
parts such that (i) the smallest part has the form ii+1 (overlined or not), (ii) the largest part is
non-overlined, (iii) the weighted differences satisfy ((mi , nj )) ≥ −1 + χ(nj overlined), (iv) r is
the number of overlined parts, andX(v)s = 1 + λ1 − c(λ1 ) +(χ(λj non-overlined) − 2c(λj )) .
j≥2

8

JEREMY LOVEJOY AND OLIVIER MALLET

Proof. We proceed more or less as in Theorem 3.1. This time, however, the term q m(m+1)/2 /(bq)mwill contribute a partition µ1 into m + 1 distinct non-negative parts with color 1 (inserting a01 ). The exponent of b again starts off as the largest part minus m. The factor 1/(q; q 2 )m+1
contributes a partition µ2 into odd parts at most 2m + 1, and this is applied to µ1 just as
before. Note that the smallest part will always have the form ii+1 . Finally, the factor (−aq)mcontributes a partition µ3 into distinct parts at most m, which is also applied as before. Sinceµ3 cannot contain 0 as a part, we never have the largest part overlined. The verification of
condition (v) is very similar to the verification of condition (iii) in Theorem 3.1.
¤
4. The combinatorial versions
We now use the discussion of the previous section to give combinatorial versions of identities(1.7) - (1.10) and (1.3) - (1.6) (in this order).
Corollary 4.1. Let A1 (k) denote the number of n-color overpartitions of k whose weighted
differences satisfy ((mi , nj )) ≥ χ(nj overlined). Let B1 (k) denote the number of overpartitionsof k into parts not divisible by 3. Then A1 (k) = B1 (k).
Proof. This follows from (1.7) and Theorem 3.1. Specifically, the summand on the left handside of (1.7) is the right hand side of (3.3) with a = b = 1 multiplied by q m(m−1)/2 . This
multiplication corresponds to adding a staircase (m − 1, m − 2, . . . , 0) to the relevant n-color
overpartitions, which simply augments the minimum weighted difference in condition (i) by 1.
Hence the left hand side of (1.7) is the generating function for A1 (k), while the right hand side
is obviously the generating function for B1 (k).
¤Corollary 4.2. Let A2 (k) denote the number of n-color overpartitions of k whose weighted differences satisfy ((mi , nj )) ≥ −1+χ(nj overlined). Let B2 (k) denote the number of overpartitionsof k into parts not divisible by 5. Then A2 (k) = B2 (k).
Proof. This follows immediately from the case a = b = 1 of Theorem 3.1 and equation (1.8). ¤Corollary 4.3. Let A3 (k) denote the number of (n + 1)-color overpartitions of k such that (i)the largest part is non-overlined, (ii) the smallest part has the form ii+1 (overlined or not), and(iii) the weighted differences satisfy ((mi , nj )) ≥ −1 + χ(nj overlined). Let B3 (k) denote thenumber of overpartitions of k whose non-overlined parts are not congruent to 0 or ±3 modulo
10. Then A3 (k) = B3 (k).
Proof. This follows immediately from equation (1.9) and the case a = b = 1 of Theorem 3.2. ¤Corollary 4.4. Let A4 (k) denote the number of (n + 1)-color overpartitions of k into nonnegative parts such that (i) the largest part is non-overlined, (ii) the smallest part has the form
ii+1 (overlined or not), and (iii) the weighted differences satisfy ((mi , nj )) ≥ χ(nj overlined).
Let B4 (k) denote the number of overpartitions of k whose non-overlined parts are not congruent
to 0 or ±1 modulo 6. Then A4 (k) = B4 (k).
Proof. This follows from equation (1.10) and the case a = b = 1 of Theorem 3.2 in the same
way that Corollary 4.1 followed from equation (1.7) and the case a = b = 1 of Theorem 3.1. ¤n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND ROGERS-RAMANUJAN IDENTITIES 9Corollary 4.5. Let A5 (k) denote the number of n-color overpartitions of k counted by A1 (k)with λ1 − c(λ1 ) even minus the number with λ1 − c(λ1 ) odd. Then
X µd¶
A5 (k) = 2.
3
d|kProof. Setting b = −1, replacing (−a)m by 2(q)m−1 , and multiplying by q m(m−1)/2 (−1)m on theright hand side of (3.3), keeping track of the corresponding weight, and then appealing to (1.3)
gives the corollary.
¤
Corollary 4.6. Let A6 (k) denote the number of n-color overpartitions of k counted by A2 (k)with λ1 − c(λ1 ) even minus the number with λ1 − c(λ1 ) odd. Then
X µd¶
A6 (k) = 2.
5d|kProof. This follows from equation (1.4) in the same way that Corollary 4.5 follows from (1.3) ¤Corollary 4.7. Let A7 (k) denote the number of (n + 1)-color overpartitions of k counted by
A3 (k) with λ1 − c(λ1 ) odd minus the number with λ1 − c(λ1 ) even. Then
X µd¶
A7 (k) =.
5d|5k+1

Proof. We replace a and b by −1 in Theorem 3.2, multiply the right hand side of (3.4) by (−1)mand keep track of the weight.
¤
Corollary 4.8. Let A8 (k) denote the number of (n + 1)-color overpartitions of k counted by
A4 (k) with λ1 − c(λ1 ) odd minus the number with λ1 − c(λ1 ) even. Then
X µd¶
A8 (k) =.
3
d|3k+1

Proof. By now, this should be routine.

¤

For an example to illustrate some of these corollaries, observe that A1 (4) = B1 (4) = 10, theoverpartitions counted by B1 (4) being(4), (4), (2, 2), (2, 2), (2, 1, 1), (2, 1, 1), (2, 1, 1), (2, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1),
while the n-color overpartitions counted by A1 (4) are(44 )(43 ), (42 ), (41 ), (44 ), (43 ), (42 ), (41 ), (31 , 11 ), (31 , 11 ).
¡ ¢
¡ ¢
¡ ¢
Notice also that A5 (4) = 2 = 2 13 + 2 23 + 2 43 , as predicted by Corollary 4.5.
For another example, we have A4 (4) = B4 (4) = 6, the overpartitions counted by B4 (4) being(4), (4), (3, 1), (3, 1), (2, 2), (2, 2),while the (n + 1)-color overpartitions counted by A4 (4) are(45 ), (41 , 01 ), (41 , 01 ), (42 , 01 ), (42 , 01 ), (43 , 01 ).
¡ ¢ ¡ ¢
Notice also that A8 (4) = 2 = 13 + 13
3 , as predicted by Corollary 4.8.

10

JEREMY LOVEJOY AND OLIVIER MALLET

5. Concluding Remarks
We wish to conclude with several remarks. First Theorem 1.1 emphasizes the point, madein [25] and based on the work in [20] and [25], that one can “sniff out” examples of q-series
with behavior resembling that of (1.1) by appealing to the combinatorics of overpartitions. The
combinatorics of ordinary partitions more typically leads to mock theta functions. For example,
the series
2X
qn(q; q 2 )n (−q)nn≥0can be interpreted in terms of ordinary n-color partitions, and it is one of Ramanujan’s 7thorder mock theta functions [11, Eq. (7.22)]
Second, some of the products in equations (1.7) - (1.10) have appeared in other studies ofoverpartitions. Namely, the product in (1.7) occurs in the case k = 3 of [24, Theorem 1.1], thecase k = 2 of [26, Theorem 1.2], Corollary 1.4 of [27], and Corollary 1.5 of [19], while the productin (1.8) occurs in the case k = 3 of [26, Theorem 1.2] and the product in (1.10) occurs in thecase k = 3 of [24, Theorem 1.2].
Finally, this is probably just the beginning for n-color overpartitions and related objects.
The generating function for unrestricted n-color overpartitions (3.2) turns up in studies of planepartitions (e.g. [21, 22, 31]) and one suspects that there is a rich theory of plane overpartitions
analogous to that of ordinary plane partitions. Additionally, the Rogers-Ramanujan-type identities in (1.7) - (1.10) can undoubtedly be embedded in infinite families of identities in the same
way as Rogers-Ramanujan-type identities for n-color partitions [8, 9]. In fact, this can probably
be further generalized to n-color overpartition pairs (c.f. [28]).
References[1] A.K. Agarwal, Partitions with N copies of N , pp.1-4, Lecture Notes in Mathematics 1234, Proceedings of
the Colloque de Combinatoire Énumerative, University of Québec at Montréal, 1985, Springer-Verlag, New
York/Berlin.
[2] A.K. Agarwal, Lattice paths and N -color partitions, Utilitas Math. 53 (1998), 71-80.
[3] A.K. Agarwal, Rogers-Ramanujan identities for n-color partitions, J. Number Theory 28 no. 3 (1988), 299305.
[4] A.K. Agarwal, New combinatorial interpretations of two analytic identities, Proc. Amer. Math. Soc. 107 no.2
(1989), 561-567.
[5] A.K. Agarwal, Identities and generating functions for certain classes of F -partitions, Ars Combin. (2000),
65-75.
[6] A.K. Agarwal, n-colour partitions, Number Theory and Discrete Mathematics (Chandigarh, 2000), 301-314,
Trends Math., Birkhäuser, Basel, 2002.
[7] A.K. Agarwal and G.E. Andrews, Rogers-Ramanujan identities for partitions with “N copies of N ”, J.
Combin. Theory Ser. A 45 (1987), 40-49.
[8] A.K. Agarwal, G.E. Andrews, and D.M. Bressoud, The Bailey lattice, J. Indian Math. Soc. 51 (1987), 57-73.
[9] A.K. Agarwal and D.M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math
136 no.2 (1989), 209-228.
[10] A.K. Agarwal, Padmavathamma, and M.V. Subbarao, Partition Theory, Atma Ram and Sons, Chandigarh,
2005.
[11] G.E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986),
113-134.
[12] G.E. Andrews, Ramanujan’s “lost” notebook V. Euler’s partition identity, Adv. Math. 61 (1986), 156-164.
[13] G.E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly 93 (1986), 708-711.

n-COLOR OVERPARTITIONS, TWISTED DIVISOR FUNCTIONS, AND ROGERS-RAMANUJAN IDENTITIES11[14] G.E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire (Salzburg, 1990), 5-18,
Publ. Inst. Rech. Math. Av. 462, Univ. Louis Pasteur, Strasbourg, 1991
[15] G.E. Andrews, Bailey chains and generalized Lambert series: I. Four identities of Ramanujan, Illinois J.
Math. 36 (1992), 251-274.
[16] G.E. Andrews, F.J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91(1988), 391-407.
[17] G.E. Andrews and D. Hickerson, Ramanujan’s “lost” notebook VII: The sixth order mock theta functions,Adv. Math. 89 (1991), 60-105.
[18] M. Baake and R.V. Moody, Similarity submodules and root systems in four dimensions, Can. J. Math. 51(1999), 1258-1276.
[19] K. Bringmann and J. Lovejoy, Rank and congruences for overpartition pairs, Int. J. Number Theory, toappear.
√[20] D. Corson, D. Favero, K. Liesinger, and S. Zubairy, Characters and q-series in Q( 2), J. Number Theory
107 (2004), 392-405.
[21] S. Corteel, O. Mallet, and C. Savelief, Plane overpartitions, in preparation.
[22] O. Foda and M. Wheeler, BKP plane partitions, preprint.
[23] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990.
[24] J. Lovejoy, Gordon’s theorem for overpartitions, J. Combin. Theory Ser. A 103 (2003), 393-401.
[25] J. Lovejoy, Overpartitions and real quadratic fields, J. Number Theory 106 (2004), 176-184.
[26] J. Lovejoy, Overpartition theorems of the Rogers-Ramanujan type, J. London Math. Soc. 69 (2004), 562-574.
[27] J. Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory 1 (2005),
215-224.
[28] J. Lovejoy and O. Mallet, Overpartition pairs and two classes of basic hypergeometric series, Adv. Math., toappear.
[29] A.V. Sills, On identities of the Rogers-Ramanujan type, Ramanujan J. 11 (2006), 403-429.
[30] L.J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. 54 (1952), 147-167.
[31] M. Vuletic, A generalization of MacMahon’s formula, preprint.
CNRS, LIAFA, Université Denis Diderot, 2, Place Jussieu, Case 7014, F-75251 Paris Cedex 05,
FRANCE
E-mail address: lovejoy@liafa.jussieu.fr
E-mail address: mallet@liafa.jussieu.fr