OVERPARTITIONSSYLVIE CORTEEL AND JEREMY LOVEJOY

1. Introduction
An overpartition of n is a partition where the first occurrence of each integer may beoverlined. For example, there are 14 overpartitions of 4,(4), (4), (3, 1), (3, 1), (3, 1), (3, 1), (2, 2), (2, 2),(2, 1, 1), (2, 1, 1), (2, 1, 1), (2, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1).
The word overpartition, a contraction of overlined partition, was coined by the first authorin [16]. Though they have been called a number of other things, like dotted partitions [8],joint partitions [55], or superpartitions [23], the term overpartitions has become standard.
Let p(n) denote the number of overpartitions of n, with the usual convention that p(0) = 1.
Since the overlined parts form a partition into distinct parts and the non-overlined parts form
an ordinary partition, we have the generating function

∞
∞ 
X
Y
1 + qnn(1.1)p(n)q =
n
1
−
q
n=0
n=1
= 1 + 2q + 4q 2 + 8q 3 + 14q 4 + 24q 5 + · · · .
Note that this generating function is the reciprocal of the classical theta function,

∞
∞ 
X
Y
1 − qnn n2(−1) q =
.
n
1
+
q
n=−∞
n=1
This fact was also noted by Hardy and Ramanujan, who anticipated the importance of thegenerating function for overpartitions in their celebrated paper that gave birth to the circle
method [32]. They wrote that (1.1) “has no very simple arithmetical interpretation; but theseries is none the less, as the direct reciprocal of simple ϑ-function, of particular interest”
and proceeded to give its asymptotics among the applications of their new method.
While combinatorial objects generated by the series in (1.1) appeared occasionally in the20th century literature (e.g. [8, 40]), it was perhaps not until a series of papers in theearly 2000s that the significance and potential of overpartitions were recognized [16, 17, 18].
In these papers, identities like the q-binomial series, Lebesgue’s identity, Ramanujan’s 1 ψ1summation, and the q-Gauss summation were given combinatorial proofs using overpartitions.
Once it became clear that overpartitions were intimately related to q-series, the floodgateswere opened. In the two subsequent decades, further q-series identities found combinatorial
proofs thanks to overpartitions [19, 29, 63, 67]; Andrews’ theory of q-difference equations forwell-poised basic hypergeometric series, which encompasses the Rogers-Ramanujan-Gordon
identities, was applied to overpartitions [6, 12, 13, 20, 21, 41, 42, 45, 46, 49]; classical partition
Date: February 28, 2024.
1

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SYLVIE CORTEEL AND JEREMY LOVEJOY

identities like Schur’s theorem were found to have overpartition versions [24, 25, 26, 27, 33, 43,
48, 56]; a theory of plane overpartitions was developed [7, 22]; Ramanujan-type congruences
and other arithmetic properties of overpartitions were studied [14, 15, 30, 36, 37, 52, 53,
58, 60, 61, 62]; the Dyson-Atkin-Swinnerton-Dyer theory of partition ranks was extended
to overpartitions [39, 44, 47, 50, 59, 66]; connections were made between overpartitions andMaass forms [9, 10, 11, 38, 54], and much, much more, including many discoveries that haveno counterpart for ordinary partitions.
Any attempt to summarize all of this in the space of a few pages is bound to fall woefully
short. We therefore limit ourselves to brief discussions of a few of the highlights – Ramanujan’s
1 ψ1 summation, overpartition ranks and mock theta functions, overpartition identities of theRogers-Ramanujan type, and plane overpartitions.
2. Combinatorics and q-series identities – Ramanujan’s 1 ψ1 summation
To illustrate the interaction between q-series identities and overpartitions, it seems fitting
to go back to one of the very first examples – Ramanujan’s 1 ψ1 summation. Using thestandard notation,Y(a; q)∞ =(1 − aq k )k≥0

and(a; q)n =(a; q)∞,(aq n ; q)∞this summation may be written(−aq; q)∞ (−bq; q)∞ X (−1/a; q)n (azq)n(−zq; q)∞ (−1/z; q)∞=,(abq; q)∞ (q; q)∞ n∈Z(−bq; q)n(b/z; q)∞ (azq; q)∞

(2.1)

1
where |b| < |z| < |aq|
and |q| < 1.
A short calculation reveals that the identity is true if and only if the constant terms in thevariable z agree on both sides. On the left hand side, the constant term is(−aq; q)∞ (−bq; q)∞,(2.2)(abq; q)∞ (q; q)∞

the generating function for pairs of overpartitions. (The variables a and b track certain
statistics, but we suppress this here.) The constant term on the right-hand side is more
subtle, but it turns out that it is the generating function for certain Frobenius symbols;
namely, two-rowed arrays
a1 a2 · · · ak,b1 b2 · · · bk
where each row is an overpartition into non-negative parts.
To prove Ramanujan’s 1 ψ1 summation, then, one needs to prove that (2.2) is the generating
function for such Frobenius symbols. This was accomplished in [17] via a combinatorial
mapping between the Frobenius symbols and pairs of overpartitions. As a bonus, the verysame mapping furnishes a combinatorial proof of the q-Gauss summation,
X (a; q)n (b; q)n (c/ab)n(c/a; q)∞ (c/b; q)∞
=
.
(c;
q)
(q;
q)
(c;
q)
(c/ab;
q)
n
n
∞
∞
n≥0

OVERPARTITIONS

3

When a, b → 0 in (2.1), one recovers Jacobi’s triple product identity,
1 X n (n+1z q 2 ) = (−1/z; q)∞ (−zq; q)∞ ,(q; q)∞ n∈Zand in this case the mapping reduces to the classical correspondence between a partition and
its Frobenius symbol.
3. Rank differences and mock theta functions
Freeman Dyson defined the rank of a partition to be the largest part minus the number ofparts, and famously conjectured that the rank could be used to refine two of Ramanujan’s
partition congruences [28],p(5n + 4) ≡ 0(mod 5),(3.1)

p(7n + 5) ≡ 0(mod 7).

(3.2)

Specifically, if N (a, m, n) denotes the number of partitions of n whose rank is congruent to amodulo m, then Dyson conjectured that for all 0 ≤ a, a′ ≤ 4 and 0 ≤ b, b′ ≤ 6 one has
N (a, 5, 5n + 4) − N (a′ , 5, 5n + 4) = 0
andN (b, 7, 7n + 5) − N (b′ , 7, 7n + 5) = 0.
Dyson’s conjecture was proven by Atkin and Swinnerton-Dyer [5], who not only showed thatthe above rank differences are 0 but also computed formulas for the generating functions forthe rank differences in every arithmetic progression 5n + d and 7n + d. One of the interesting
aspects of these generating functions is that a number of them are related to Ramanujan’s
5th or 7th order mock theta functions. To give just one example [35],X(N (0, 7, 7n) − N (2, 7, 7n)) q n = F0 (q),
n≥0

where
2

qn
F0 (q) =(q n+1 ; q)nn≥0Xis one of Ramanujan’s 7th order mock theta functions.
In the case of overpartitions, the methods of Atkin and Swinnerton-Dyer can be adapted toprove formulas for the generating functions for rank differences [39, 50, 59]. While there are no
simple congruences like (3.1) or (3.2) for p(n), the rank differences still have nice generating
functions and connections to mock theta functions. For example, if N (a, m, n) denotes thenumber of overpartitions of n with rank congruent to a modulo m, we have [51, 59]X
N (0, 5, 5n + 1) − N (2, 5, 5n + 1) q n = 2ϕ(q)
n≥0

and
X
n≥0


N (1, 6, 3n + 2) − N (3, 6, 3n + 2) q n = 2ω(q),

4

where

SYLVIE CORTEEL AND JEREMY LOVEJOY

n+1
q( 2 )
ϕ(q) =(q; q 2 )n+1
n≥0

X

is one of Ramanujan’s 10th order mock theta functions andX q 2n2 +2nω(q) =(q; q 2 )2n+1
n≥0
is one of Ramanujan’s 3rd order mock theta functions.
The most striking results arise, however, when considering the overpartition rank differences
modulo 2. Here the rank differences turn out to be related to the Hurwitz class numbers
H(n) [10]. We recall that H(n) is the number of equivalence classes of positive definite
binary quadratic forms of discriminant −n, with classes containing a multiple of x2 + y 2 orx2 +xy +y 2 counted with weight 1/2 and 1/3, respectively. The generating function for H(n),X
1
H(n)q n ,H(q) = − +12 n≥1is known as the Zagier-Eisenstein series. This was perhaps the first example of a mock
modular form that explicitly appeared in the literature [65], long before the notion wasformalized. Specifically, it is a weight 3/2 mock modular form with shadow proportional tothe classical theta function
X 2
qn .
Θ(q) =n∈Z

It turns out that the same is true for the generating function for (−1)n (N (0, 2, n)−N (1, 2, n)),and this gives rise to the identityX1
N (0, 2, n) − N (1, 2, n) (−1)n q n = −16H(q) − Θ3 (q).
3n≥0Since the coefficients of Θ3 (q) can also be expressed in terms of H(n), this leads to resultslike
N (0, 2, 8n + 3) − N (1, 2, 8n + 3) = 24H(8n + 3).
This is all in stark contrast to the case of partitions, where the generating function for
N (0, 2, n) − N (1, 2, n) is Ramanujan’s 3rd order mock theta function f (q).
4. Overpartition identities
Since Euler discovered that the number of partitions of n into distinct parts is equal to
the number of partitions of n into odd parts, the search for such identities has been oneof the principal themes in the theory of partitions. The most famous of them all are theRogers-Ramanujan identities, which correspond to the q-series identities of the same name,
X q n2 +(2−i)n1= 3−i 5.
(q; q)n(q ; q )∞ (q 2+i ; q 5 )∞n≥0Here i = 1 or 2. In terms of partitions, the Rogers-Ramanujan identities state that the
number of partitions of n into parts differing by at least two with at most i − 1 occurrencesof 1 is equal to the number of partitions into parts not congruent to 0, ±i modulo 5.

OVERPARTITIONS

5The Rogers-Ramanujan identities were famously generalized by Gordon [31], whose result
may be stated as follows: For k ≥ 2 and 1 ≤ i ≤ k, let Ak,i (n) denote the number of partitionsof n where at most i − 1 of the parts are equal to 1 and the total number of occurrences ofj and j + 1 together is at most k − 1 for any integer j, and let Bk,i (n) denote the number ofpartitions of n into parts not congruent to 0, ±i modulo 2k + 1. Then
Ak,i (n) = Bk,i (n).
Note that the case k = 2 recovers the Rogers-Ramanujan identities.
Andrews showed how Gordon’s result could be obtained from an analysis of certain wellpoised q-hypergeometric series [1],X (−a)n q kn2 +n−in xkn (1 − xi q 2ni )(−1/a; q)n (−axq n+1 ; q)∞
Hk,i (a; x; q) =(q; q)n (xq n ; q)∞n≥0
andJk,i (a; x; q) = Hk,i (a; xq; q) + axqHk,i−1 (a; xq; q).
It turns out that these series (i) satisfy simple q-difference equations and (ii) become infinite
products under various specializations of the parameters. This not only leads to the RogersRamanujan-Gordon identities, but a large number of other identities of this type [4, Chapter
7].
The series Hk,i (a; x; q) and Jk,i (a; x; q) and their q-difference equations apply naturally to
overpartitions as well. The second author [41] and Chen-Sang-Shi [12] have used this to findan overpartition version of the Rogers-Ramanujan-Gordon theorem. For brevity we record
only the case i = k: Let Ak,k (n) denote the number of overpartitions of n where parts occur
at most k − 1 times, and where the total number of occurrences of j and j + 1 together is at
most k if j occurs overlined and at most k − 1 otherwise. Let B k,k (n) denote the number ofoverpartitions of n into parts not divisible by k. Then
Ak,k (n) = B k,k (n).
Another famous partition identity is due to Schur [57]. One version of Schur’s theorem
says that if S(n) denotes the number of partitions of n into distinct parts not divisible by 3
and T (n) denotes the number of partitions of n where consecutive parts differ by at least 3
with multiples of 3 differing by at least 6, thenS(n) = T (n).
In [43], the second author found an overpartition version of Schur’s theorem. It says that if
S(k, n) denotes the number of overpartitions of n into parts not divisible by 3, k of which are
non-overlined, and T (k, n) denotes the number of overpartitions of n with k non-overlined
parts where consecutive parts differ by at least 3 if the smaller is overlined or both parts aredivisible by 3, and by at least 6 if the smaller is overlined and both parts are divisible by 3,thenS(k, n) = T (k, n).
Note that this is a “perfect” generalization of Schur’s partition theorem, in the sense that
if there are no non-overlined parts then we recover Schur’s result. Schur’s theorem was
generalized in two different ways by Andrews, from the modulus 3 to the modulus 2k − 1[2, 3], and Andrews’ theorems were both generalized to overpartitions by Dousse [25, 26].

6

SYLVIE CORTEEL AND JEREMY LOVEJOY

5. Plane overpartitions
A plane partition is a two-dimensional array of nonnegative integers where each row andeach column is a partition. An example of a plane partition Π is:4 4 4 34 4 3 13 3
3Plane partitions generalize ordinary partitions. Indeed a plane partition with one row is apartition. We denote by |Π| the sum of the entries of the plane partition. In the example
above |Π| = 36. The generating function of plane partitions was computed by MacMahon :X
Y  1 i
|Π|
q =
.
i
1
−
q
i≥1
Π a plane
partition

A plane partition may be represented visually by the placement of a stack of unit cubes in
the corner of a room or as a rhombus tiling.
A plane overpartition [22] is a filling of a Ferrers diagram where (i) each row is an overpartition where the final occurrence of an integer may be overlined, (ii) each column is an
overpartition where the first occurrence of an integer can be overlined or not and all the otheroccurrences of this integer are overlined. An example of a plane overpartition Π is:4 4 4̄ 3̄
4̄ 3 3 3̄
4̄ 3̄
3
We denote by |Π| the sum of the entries of the plane overpartition and by o(Π) the numberof overlined entries. In the example above |Π| = 38 and o(Π) = 6.
The generating function of plane overpartitions isXY(1 + aq i )iao(Π) q |Π| =.
(1 − q i )⌈i/2⌉ (1 − a2 q i )⌊i/2⌋
i≥1
Π a plane
overpartition

If a = 1, one can think of this as the overpartition analogue of the generating function ofplane partitions,X
Y (1 + q i )iq |Π| =
(1 − q i )ii≥1
Π a plane
overpartition

However, if no entry is overlined (i.e. if a = 0), we recover a plane partition where all
the columns are strict. These objects are naturally related to Schur functions, while plane
overpartitions are related to P and Q-Schur functions and super Schur functions.
Another natural analogue of a plane partition is called a pyramid partition [64]. Pyramid
partitions are infinite stacks of square bricks in the shape of a pyramid where a finite numberof square bricks have been removed. These are in bijection with pairs of plane overpartitions

OVERPARTITIONS

7of the same shape and arise naturally from a computation of Donaldson-Thomas invariantsof orbifolds. Their generating function is :Y (1 + aq 2i−1 )2i−1.
2i )i (1 − a2 q 2i )i
(1
−
q
i≥1
When a = 0 and q is replaced by its square root, we recover the generating function of planepartitions. Both plane overpartitions and pyramid partitions are in bijection with certain
domino tilings [7, 22].
6. Conclusion
Overpartitions are to be found wherever partitions play a role. It is now common practiceto ask “What about the overpartition case?” when one proves a theorem about partitions.
This is often much more than a simple exercise involving a routine generalization. Sometimes
even discovering the right definition can be a conundrum. And as we have seen in Section 3,
sometimes the overpartition case holds an unexpected surprise. Moreover, overpartitions canbe crucial to understanding partitions. The best example of this is a recent proof of a very
general partition identity conjectured by Bressoud, where overpartitions played a key role [34].
As the authors wrote, “It should be stressed that the overpartition analogues considered inthis paper are not merely a matter of extension and specialization. In fact, they play anessential role and serve as an indispensable structure in tackling the conjecture of Bressoud
formulated in terms of ordinary partitions.”
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Department of Mathematics, University of California Berkeley, Evans Hall, Berkeley,
CA 94720
Email address: corteel@berkeley.edu
CNRS, Université Paris Cité, Bâtiment Sophie Germain, Case courier 7014, 8 Place Aurélie
Nemours. 75205 Paris Cedex 13, FRANCE
Email address: lovejoy@math.cnrs.fr