PARTITIONS AND OVERPARTITIONS WITH ATTACHED PARTSJEREMY LOVEJOYAbstract. We show how to interpret a certain q-series as a generating function for overpartitions with attached parts. A number of families of partition theorems follow as corollaries.

1. Introduction
The purpose of this paper is to unify and generalize some families of partition theorems,including Gordon’s generalization of the Rogers-Ramanujan identities [9], Andrews’ generalization of the Göllnitz-Gordon identities [1], the two Gordon’s theorems for overpartitions [10], theAndrews-Santos identities for partitions with attached odd parts [3], and their overpartition analogue [11]. This will be accomplished through a combinatorial study of the series Gk,i (a, b; x; q),defined by∞(−bxq)∞ X xkn q n((2k−1)n+3)/2+(k−i)n (xq)n (1/a, −1/b)n (ab)nGk,i (a, b; x; q) =(1.1)(xq)∞(q, axq, −bxq)nn=0!
abxi q (2n+1)i−2n (1 − q n /a)(1 + q n /b).
×1+(1 − axq n+1 )(1 + bxq n+1 )Here we have employed the usual q-series notation [8]. In the context of Andrews’ theory ofwell-poised basic hypergeometric series [2], the series Gk,i (a, b; x; q) can be expressed in terms of
his Jk,i (a, b; x; q)2 as
Jk,i (1/a, −1/b; x; q)2
Gk,i (a, b; x; q) =.
(1.2)(axq)∞
Our main result is an interpretation of the coefficient of ar bs xt q n in Gk,i (a, b; x; q) using theframework of overpartitions and overpartition pairs [5, 12]. Recall that by an overpartition wemean a partition in which the first occurrence of a number may be overlined. We employ thenotation fj (λ) to denote the number of occurrences of j in λ.
Theorem 1.1. For k ≥ 2 and 1 ≤ i ≤ k, let gk,i (r, s, t, n) denote the number of overpartitionpairs (λ, µ) of n with k − i non-overlined zeros in λ and t positive parts, r of which are in µ, ands of which are overlined, where for each j ≥ 0 we have (i) fj (λ) + fj+1 (λ) + fj+1 (λ) ≤ k − 1,(ii) if fj (λ) + fj+1 (λ) + fj+1 (λ) = k − 1, then j + 1 may occur in µ, and (iii) if fj (λ) = k − 1,then j + 1 may occur in µ. ThenXGk,i (a, b; x; q) =gk,i (r, s, t, n)ar bs xt q n .
(1.3)
r,s,t,n≥0Date: October 19, 2007.
2000 Mathematics Subject Classification. 11P81, 05A17, 33D15.
12JEREMY LOVEJOY

Following [3], we say that the parts of the overpartition µ are attached to the overpartition λ.
We might make a few remarks here. First, the k − i “phantom” zeros in the statement ofTheorem 1.1 are just an artifice to streamline the definition of the gk,i (r, s, t, n). Second, one
may compare this theorem, where achieving a boundary condition affords more freedom to thecombinatorial objects under consideration, to results like those in [4, 6] or [12], where achieving
a boundary condition induces further restrictions on the objects. Finally, we note that the casea = 0 of Theorem 1.1 is Corollary 1.1 of [7].
In the next section we use q-difference equations to establish Theorem 1.1. In Section 3 wemention how the various results listed in the opening paragraph are corollaries of Theorem 1.1
and offer another example of a family of overpartition identities that can be deduced from thistheorem:
Theorem 1.2. For all k ≥ 2 and 2 ≤ i ≤ k, let Hk,i (n) denote the number of overpartitions νof n having k − i phantom zeros such that (i) for each j ≥ 0, f2j (ν) + f2j+1 (ν) + f2j+2 (ν) ≤ k − 1,(ii) if f2j (ν) + f2j+1 (ν) + f2j+2 (ν) = k − 1, then 2j + 1 may occur (without restriction), and′ (n) denote the number of overpartitions of(iii) if f2j (ν) = k − 1, then 2j may occur. Let Hk,in where the non-overlined parts are even and not divisible by 4k − 2 and the overlined parts are′ (n).
odd or congruent to ±(2i − 2) modulo 4k − 2. Then Hk,i (n) = Hk,i2. Proof of Theorem 1.1
Henceforth we assume that k ≥ 2 and that 1 ≤ i ≤ k. Using equation (1.2) and the qdifference equations for the Jk,i (a, b; x; q)2 (and related functions) recorded in [2, Eq. 2.1, 2.2,2.3], we may deduce the following q-difference equations for the Gk,i (a, b; x; q):
Lemma 2.1.
Gk,1 (a, b; x; q) =(1 + abxq)
Gk,k (a, b; xq; q),(1 − axq)

Gk,2 (a, b; x; q) = Gk,k (a, b; xq; q)bxq+
Gk,k (a, b; xq; q)
1 − axq
xq+
Gk,k−1 (a, b; xq; q),
1 − axq

(2.1)
(2.2)

and, for i ≥ 3,Gk,i (a, b; x; q) − Gk,i−1 (a, b; x; q) =−+−(xq)i−1
Gk,k−i+1 (a, b; xq; q)
1 − axqa(xq)i−1
Gk,k−i+2 (a, b; xq; q)
1 − axqb(xq)i−1
Gk,k−i+2 (a, b; xq; q)
1 − axqab(xq)i−1
Gk,k−i+3 (a, b; xq; q).
1 − axq

(2.3)

(2.4)

PARTITIONS AND OVERPARTITIONS WITH ATTACHED PARTS

3The presence of the minus sign in this final equation is rather undesirable from a combinatorial
point of view, but fortunately there is a nicer set of equations which follow by induction from
Lemma 2.1.
Lemma 2.2.
Gk,1 (a, b; x; q) =(1 + abxq)
Gk,k (a, b; xq; q),(1 − axq)(2.5)

and, for i ≥ 2,(xq)i−1
Gk,k−i+1 (a, b; xq; q)
1 − axqb(xq)i−1+Gk,k−i+2 (a, b; xq; q)
1 − axq
i−3
X+b(xq)v+1 Gk,k−v (a, b; xq; q)Gk,i (a, b; x; q) =+

v=0
i−2X

(2.6)

(xq)v Gk,k−v (a, b; xq; q).

v=0

Since

Gk,i (a, b; 0; q) = 1(2.7)
for all k and i, equations (2.5) and (2.6) uniquely define the functions Gk,i (a, b; x; q). Let
Xb k,i (a, b; x; q) =G
gk,i (r, s, t, n)ar bs xt q n ,(2.8)
r,s,t,n≥0with the gk,i (r, s, t, n) defined as in Theorem 1.1. To prove the theorem we will show thatb k,i (a, b; x; q) satisfy the same q-difference equations as the Gk,i (a, b; x; q). The boundary
the Gcondition (2.7) is clear since there is only one overpartition pair having no parts, the empty one.
b k,1 (a, b; x; q) is the generating function for overpartition pairs (λ, µ) where λThe function G
has no ones and at most k − 1 twos, and µ has ones without restriction. Subtracting one fromeach part ≥ 2, we see thatb k,k (a, b; xq; q).
b k,1 (a, b; x; q) = (1 + abxq) GG(1 − axq)(2.9)

This is (2.5).
Establishing (2.6) is a bit more involved. For i ≥ 2, we divide those overpartition pairs (λ, µ)b k,i (a, b; x; q) into four groups, depending on whether f1 (λ) + f (λ) = i − 1 andgenerated by G
1
whether 1 occurs in λ. First, if f1 (λ) + f1 (λ) = i − 1 and 1 appears in λ, then there may be up tok − i + 1 twos in λ and 1 (but not 1) can appear without restriction in µ. Hence by subtracting
one from each part ≥ 2 we see that these overpartition pairs are generated bybxq(xq)i−2 b
Gk,k−i+2 (a, b; xq; q).
(1 − axq)(2.10)

Next, if f1 (λ) + f1 (λ) = i − 1 and 1 does not appear in λ, then there may be up to k − i twosin λ and 1 (but not 1) can appear without restriction in µ. Hence by subtracting one from each

4JEREMY LOVEJOY

part ≥ 2 we see that these overpartition pairs are generated by(xq)i−1 b
Gk,k−i+1 (a, b; xq; q).
(1 − axq)(2.11)

Now if f1 (λ) + f1 (λ) < i − 1 and 1 appears in λ, then there are no ones in µ. Supposing that
1 occurs v times in λ, 0 ≤ v ≤ i − 3, then there are at most k − v − 1 twos in λ. Hence theseoverpartition pairs are generated byi−3X
v=0b k,k−v (a, b; xq; q).
bxq(xq)v G(2.12)

Finally, if f1 (λ) + f1 (λ) < i − 1 and 1 does not appear in λ, then there are still no ones in µ.
Supposing that 1 occurs v times in λ, now with 0 ≤ v ≤ i − 2, then there are at most k − v − 1
twos in λ. Hence these overpartition pairs are generated byi−2
X
v=0b k,k−v (a, b; xq; q).
(xq)v G(2.13)

b k,i (a, b; x; q) satisfy (2.6) for i ≥ 2,Putting these four cases together shows that the functions G
b k,i (a, b; x; q), completing the proof of Theorem
and we may now conclude that Gk,i (a, b; x; q) = G
1.1.
3. The Corollaries

From (1.1) and the fact that(a)−n =(−1)n q n(n+1)/2,an (q/a)n

it follows that
Gk,i (a, b; 1; q) =(−bq)∞ X q n((2k−1)n+3)/2+(k−i)n (1/a, −1/b)n (ab)n.
(q)∞(aq, −bq)n(3.1)

n∈Z

By employing Jacobi’s triple product identity [8],X2z n q n = (−zq, −q/z, q 2 ; q 2 )∞ ,

(3.2)

n∈Z

we find that these Gk,i (a, b; 1; q) are often infinite products with nice partition-theoretic interpretations. For example, when q = q 2 and a = b = 1/q, then this series becomes the product(−q 2i−2 , −q 4k−2i , q 4k−2 ; q 4k−2 )∞ (−q; q 2 )∞,(q 2 ; q 2 )∞(3.3)

′ (n) of Theorem 1.2.
which is clearly the generating function for the Hk,iOn the other hand, we may also interpret Gk,i (1/q, 1/q; 1; q 2 ) using Theorem 1.1. In this case,starting with an overpartition pair (λ, µ) counted by gk,i (r, s, t, n), the non-overlined (positive)
parts j of λ become 2j, the overlined parts j of λ become 2j − 1, the non-overlined parts j of µ
become 2j − 1, and the overlined parts of µ become 2j − 2. Hence we may regard the resulting
overpartition pair (λ, µ) as just an overpartition ν, and the conditions (i) − (iii) on (λ, µ) inPARTITIONS AND OVERPARTITIONS WITH ATTACHED PARTS

5Theorem 1.1 transform into the conditions (i) − (iii) on ν in Theorem 1.2. In other words,Gk,i (1/q, 1/q; 1; q 2 ) is the generating function for Hk,i (n). This establishes Theorem 1.2
The classes of theorems in the first paragraph of the introduction may be deduced in exactly
the same way. They correspond to Gk,i (0, 0; 1; q), Gk,i (0, 1/q; 1; q 2 ), Gk,k (0, 1; 1; q), Gk,1 (0, 1/q; 1; q),Gk,i (1/q, 0; 1; q 2 ), and Gk,k (1/q, 1; 1; q 2 ), respectively. The details are left to the reader.
Corollary 3.1 (Gordon, [9]). Let Ak,i (n) denote the number of partitions λ of n with k − izeros, such that for all j ≥ 0 we have fj (λ) + fj+1 (λ) ≤ k − 1. Let A′k,i (n) denote the number ofpartitions of n whose parts are not congruent to 0 or ±i modulo 2k + 1. Then Ak,i (n) = A′k,i (n).
Corollary 3.2 (Andrews, [1]). Let Bk,i (n) denote the number of partitions of λ of n with k − i′ (n) denotezeros, such that for all j ≥ 0 we have f2j (λ) + f2j+1 (λ) + f2j+2 (λ) ≤ k − 1. Let Bk,ithe number of partitions of n where parts are neither congruent to 2 modulo 4 nor congruent to′ (n).
0 or ±(2i − 1) modulo 4k. Then Bk,i (n) = Bk,iCorollary 3.3 (Lovejoy, [10]). Let Ck (n) denote the number of overpartitions λ of n with nozeros, such that for all j ≥ 1 we have fj (λ) + fj+1 (λ) + fj+1 (λ) ≤ k − 1. Let Ck′ (n) denote thenumber of overpartitions of n whose parts are not divisible by k. Then Ck (n) = Ck′ (n).
Corollary 3.4 (Lovejoy, [10]). Let Dk (n) denote the number of overpartitions λ of n where 1
does not occur (non-overlined) and for all j ≥ 1 we have fj (λ) + fj (λ) + fj+1 (λ) ≤ k − 1. Let
Dk′ (n) denote the number of overpartitions of n whose non-overlined parts are not congruent to0 or ±1 modulo 2k. Then Dk (n) = Dk′ (n).
Corollary 3.5 (Andrews-Santos, [3]). Let Ek,i (n) denote the number of partitions λ of n having
k − i zeros, such that for all j ≥ 0 we have (i) f2j (λ) + f2j+2 (λ) ≤ k − 1 and (ii) if f2j (λ) +′ (n) denote the number
f2j+2 (λ) = k − 1, then 2j + 1 may occur without restriction. Let Ek,iof partitions of n into parts that are either even and not divisible by 4k or odd, distinct, and′ (n).
congruent to ±(2i − 1) modulo 4k. Then Ek,i (n) = Ek,iCorollary 3.6 (Lovejoy, [11]). Let Fk (n) denote the number of overpartitions λ of n withno zeros, such that for all j ≥ 0 we have (i) f2j (λ) + f2j+2 (λ) + f2j+2 (λ) ≤ k − 1, (ii) if
f2j (λ)+f2j+2 (λ)+f2j+2 (λ) = k−1, then 2j+1 may occur (non-overlined and without restriction),and (iii) if f2j (λ) = k −1, then 2j + 1 may occur. Let Fk′ (n) denote the number of overpartitionsof n into parts that are either even and not divisible by 4k − 2 or odd and congruent to 2k − 1
modulo 4k − 2. Then Fk (n) = Fk′ (n).
4. Concluding Remarks
The main point of this paper is that by taking a broad look at some of the basic hypergeometricseries commonly employed in the proofs of partition identities, one is led to general theorems
which unify many of these identities and easily yield further results. It should also not be
overlooked that the framework of overpartitions is rather useful in this endeavor.
As previously mentioned, the case a = 0 of Theorem 1.1 was already treated in [7]. In fact, theauthors related a number of different objects to the generating function Gk,i (0, b; 1; q), such as
lattice paths, overpartitions with bounded successive ranks, and overpartitions with a specified
Durfee decomposition. One wonders if all of these concepts can be extended to the generalGk,i (a, b; x; q) studied here.

6JEREMY LOVEJOY

References[1] G.E. Andrews, A generalization of the Göllnitz-Gordon partition identities, Proc. Amer. Math. Soc. 8 (1967),945–952.
[2] G.E. Andrews, On q-difference equations for certain well-poised basic hypergeometric series, Quart. J. Math.
Ser. 2 19 (1968), 433-447.
[3] G.E. Andrews and J.P.O. Santos, Rogers-Ramanujan type identities for partitions with attached odd parts,Ramanujan J. 1 (1997), 91-99.
[4] D.M. Bressoud, A generalization of the Rogers–Ramanujan identities for all moduli, J. Combin. Theory Ser.
A 27 (1979), 64–68.
[5] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623-1635.
[6] S. Corteel, J. Lovejoy, and O. Mallet, An extension to overpartitions of the Rogers-Ramanujan identities foreven moduli, preprint.
[7] S. Corteel and O. Mallet, Overpartitions, lattice paths, and Rogers-Ramanujan identities, preprint[8] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990.
[9] B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961),
393-399.
[10] J. Lovejoy, Gordon’s theorem for overpartitions, J. Comb. Theory Ser. A. 103 (2003), 393-401.
[11] J. Lovejoy, Overpartition theorems of the Rogers-Ramanujan type, J. London Math. Soc. 69 (2004), 562-574.
[12] J. Lovejoy, Overpartition pairs, Ann. Inst. Fourier 56 (2006), 781-794.
CNRS, LIAFA, Université Denis Diderot, 2, Place Jussieu, Case 7014, F-75251 Paris Cedex 05,
FRANCE
E-mail address: lovejoy@liafa.jussieu.fr