PARTITIONS WITH ROUNDED OCCURRENCES AND ATTACHED PARTSJEREMY LOVEJOY
To George Andrews, with admiration, on the occasion of his 70th birthday
Abstract. We introduce the number of (k, i)-rounded occurrences of a part in a partition anduse q-difference equations to interpret a certain q-series Sk,i (a; x; q) as the generating function
for partitions with bounded (k, i)-rounded occurrences and attached parts. When a = 0 thesepartitions are the same as those studied by Bressoud in his extension of the Rogers-RamanujanGordon identities to even moduli. When a = 1/q we obtain a new family of partition identities.

1. Introduction and statement of results
In 1968, greatly generalizing work of Rogers [25] and Selberg [26], Andrews defined a family ofbasic hypergeometric series Jλ,k,i (a1 , a2 , . . . , aλ ; x; q) and established q-difference equations involving them [5]. This work became one of the foundations of modern partition theory. Andrews
had already seen how to use some of these q-difference equations to prove families of partition
identities [1, 2, 3, 4], including Gordon’s combinatorial generalization of the Rogers-Ramanujan
identities, and over the next decade many further partition identities [6, 8, 9, 14, 16] would be
deduced from the Jλ,k,i (a1 , a2 , . . . , aλ ; x; q) and their q-difference equations.
With the focus on analytic identities, motivated in large part by the burgeoning applications
in statistical mechanics [10] and the advent of the powerful Bailey pair method [11], the 80’s
and 90’s saw the study of q-difference equations fall out of favor. Over the last decade or so,
however, a string of papers have shown that there is still much to be discovered in Andrews’
Jλ,k,i (a1 , a2 , . . . , aλ ; x; q) [12, 13, 18, 20, 21, 22, 23, 24]. The present work is yet another contribution to this list.
We study the series Sk,i (a; x; q), defined for k ≥ 2 and 1 ≤ i ≤ k using the usual q-series
notation [19] by2

Sk,i (a; x; q) :=1 X an x(k−1)n q (k−1)n +(k−i+1)n (x2 q 2 , 1/a; q 2 )n(xq)∞(q 2 , ax2 q 2 ; q 2 )nn≥0!
axi q (2n+1)i−2n (1 − q 2n /a)× 1+.
(1 − ax2 q 2n+2 )(1.1)

In terms of Andrews’ series, we have
Sk,i (a; x; q) :=(−xq)∞
J k−1 i (1/a; x2 ; q 2 ).
(ax2 q 2 ; q 2 )∞ 1, 2 , 2

Date: July 4, 2009.
1(1.2)

2JEREMY LOVEJOY

We will describe the coefficient of at xm q n of Sk,i (a; x; q) in terms of partition pairs, using thenumber of (k, i)-rounded occurrences of a part j in a partition λ.
Definition 1.1. Denote by fj (λ) the number of occurrences of j in λ. The number of (k, i)(k,i)
rounded occurrences of a part j in a partition λ, denoted fj (λ), is defined using the usual
charactersitic function by(k,i)

f2j (λ) := f2j (λ) + χ(f2j 6≡ (k − i) (mod 2)),

(1.3)

(k,i)

(1.4)

f2j+1 (λ) := f2j+1 (λ) + χ(f2j+1 6≡ (i − 1)(mod 2)).

Notice that with this definition we require nothing about the parity of the number of occurrences of a part, only that the number of occurrences be regarded as having a certain
parity by rounding up, if necessary. To illustrate the definition, consider the partition λ =(4,3)
(4,2)
(5,2)
(4,4)
(6, 6, 6, 4, 4, 3, 3, 3, 3, 1). Then we have, for example, f1= 2, f2= 0, f3= 5, f4= 2,(4,4)(5,3)
f5= 1, and f6= 4.
We now define the partitions pairs of interest.
Definition 1.2. Let sk,i (n) denote the number of partition pairs (λ, µ) of n such that:(k,i)(i) f1(λ) ≤ i − 1,(k,i)(ii) if f1(λ) = i − 1 then 1 may occur an even number of times in µ,(iii) if i = 1 then 1 may occur unrestricted in µ,(k,i)(iv) for each j ≥ 1 we have fj(k,i)(v) for each j ≥ 1, if fj
times in µ,(k,i)(vi) for each j ≥ 1, if fj(k,i)(λ) + fj+1 (λ) ≤ k − 1,(k,i)(λ) + fj+1 (λ) = k − 1, then j + 1 may occur an even number of(λ) = k − 1, then j + 1 may occur unrestricted in µ.

We are now ready to state the main theorem.
Theorem 1.3. Let sk,i (t, m, n) denote the number of partition pairs counted by sk,i (n) such thatP f (µ)
Pm = j (fj (λ) + fj (µ)) and t = j ⌈ j 2 ⌉. ThenXsk,i (t, m, n)at xm q n = Sk,i (a; x; q).
(1.5)
t,m,n≥0Theorem 1.3 shows that special cases of the functions Sk,i (a; x; q) are generating functions forsome well-known partitions. For example, a few moments’ consideration (or, to bypass Theorem
1.3, an appeal to (1.1)) reveals that ordinary partitions are generated by Sk,i (1; x; q) (for any kand i). It is also not hard to see that partitions into distinct parts are generated by S3,2 (0; 1; q).
More generally, the partitions generated by Sk,i (0; x; q) may be identitifed with those studied by
Bressoud in his extension to even moduli of Gordon’s generalization of the Rogers-Ramanujan
identities (i.e., the partitions counted by bk,i (m, n) in [14]). Setting a = 0 and x = 1 in (1.1)

PARTITIONS WITH ROUNDED OCCURRENCES AND ATTACHED PARTS

and appealing to the triple product identity [19, p.239, Eq. (II.28)],
X
2z n q n = (−q/z, −zq, q 2 ; q 2 )∞ ,3(1.6)

n∈Z

we then recover Bressoud’s result [14, Theorem, j = 0] in the following form:
Corollary 1.4 (Bressoud, [14]). For k ≥ 2 and 1 ≤ i < k, let Bk,i (n) denote the number ofpartitions λ of n such that:(k,i)(i) f1(λ) ≤ i − 1,(k,i)(ii) for each j ≥ 1 we have fj(k,i)(λ) + fj+1 (λ) ≤ k − 1.

Let Ak,i (n) denote the number of partitions of n into parts not congruent to 0 or ±i modulo 2k.
Then Ak,i (n) = Bk,i (n).
Another interesting consequence of Theorem 1.3 arises when we set a = 1/q. It is convenient
to state this result in terms of overpartitions, which are partitions in which the first occurrenceof a part may be overlined.
Corollary 1.5. For k ≥ 2 and 1 < i < k, let Bk,i (n) denote the number of overpartition pairs(λ, µ) of n such that:(i) λ is an ordinary partition counted by Bk,i (n) (see Corollary 1.4),(k,i)(ii) if f1(λ) = i − 1, then 1 may occur (non-overlined and unrestricted) in µ,
(k,i)

(k,i)

(iii) for j ≥ 1, if fj (λ) + fj+1 (λ) = k − 1, then 2j + 1 may occur (non-overlined andunrestricted) in µ,(k,i)(iv) for j ≥ 1 if fj(λ) = k − 1 then j may appear in µ.

Let Ak,i (n) denote the number of overpartitions of n where non-overlined parts are not divisibleby 2k − 2 and overlined parts are ±(i − 1) (mod 2k − 2). Then Ak,i (n) = Bk,i (n).
Despite the requirement that 1 < i < k above, there is still an identity when i = 1 or k.
Indeed, the proof of Corollary 1.5 presented in Section 2 applies equally well when i = 1 ork. The definition of Bk,i (n) is still valid (with a suitable modification for i = 1 arising fromcondition (iii) in Definition 1.2), and the generating functions for Ak,1 (n) and Ak,k (n) areAk,1 (n) =2(−q 2k−2 ; q 2k−2 )2∞ (q 2k−2 ; q 2k−2 )∞(q)∞

Ak,k (n) =(−q k−1 ; q 2k−2 )2∞ (q 2k−2 ; q 2k−2 )∞.
(q)∞

andWe let the reader interpret these products as he pleases.

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JEREMY LOVEJOY

2. Proofs of Theorem 1.3 and Corollary 1.5
Using (1.2) and [5, Theorem 1] one may compute that
(1 + axq)
Sk,k (a; xq; q),(1 − ax2 q 2 )(2.1)(1 + xq)
Sk,k−1 (a; xq; q),(1 − ax2 q 2 )(2.2)

Sk,1 (a; x; q) =Sk,2 (a; x; q) =and for 3 ≤ i ≤ k,(xq)i−2 (1 + xq)
Sk,k−i+1 (a; xq; q)(1 − ax2 q 2 )(2.3)
a(xq)i−2 (1 + xq)
Sk,k−i+3 (a; xq; q).
−(1 − ax2 q 2 )The final q-difference equation is not terribly useful combinatorially. However, there is anotherq-difference equation which may be easily deduced from (2.1), (2.2), and (2.3) using induction(equation (2.3) providing the induction step). This method of eliminating the minus sign isinspired by [13].
Sk,i (a; x; q) − Sk,i−2 (a; x; q) =Lemma 2.1. If i ≥ 2 is even thenSk,i (a; x; q) =(xq)i−2 (1 + xq)
Sk,k−i+1 (a; xq; q)(1 − ax2 q 2 )(2.4)(i−2)/2+X

2v−2(xq)(1 + xq)Sk,k−2v+1 (a; xq; q),v=1and if i ≥ 3 is odd thenSk,i (a; x; q) = Sk,k (a; xq; q) +(xq)i−2 (1 + xq)
Sk,k−i+1 (a; xq; q)(1 − ax2 q 2 )(2.5)(i−3)/2+X

2v−1(xq)(1 + xq)Sk,k−2v (a; xq; q).

v=1

Proof of Theorem 1.3. Notice that together with the initial condition Sk,i (a; 0; q) = 1, the qdifference equations (2.1), (2.4), and (2.5) uniquely define the functions Sk,i (a; x; q). To prove
Theorem 1.3 then, we define
XSbk,i (a; x; q) :=sk,i (t, m, n)at xm q nt,m,n≥0

and show that the Sbk,i (a; x; q) satisfy the same defining conditions. That Sbk,i (a; 0; q) = 1 follows
from the fact that the only partition without any parts whatsoever is the empty partition of 0.
We now turn to (2.1). Let (λ, µ) be a partition pair counted by Sbk,1 (a; x; q). By definition, we(k,1)
have f1 (λ) = 0, f2 (λ) ≤ k − 1, and f1 (µ) is unrestricted. Removing the 1’s and subtracting
one from each part ≥ 2, we see that(1 + axq) b
Sbk,1 (a; x; q) =Sk,k (a; xq; q).
(1 − ax2 q 2 )PARTITIONS WITH ROUNDED OCCURRENCES AND ATTACHED PARTS

5(Notice that for (k, i) = (k, 1) and (k, k), the residue classes modulo 2 of (k − i) and (i − 1)
are interchanged, so that subtracting one from each part is consistent with the definition of thenumber of (k, i)-rounded occurrences in Definition 1.1 and the conditions on the sk,i (t, m, n) in
Theorem 1.3. This will be the case throughout the proof, though we shall not mention it again.)
Next we treat (2.4). Suppose that (λ, µ) is a partition pair counted by Sbk,i (a; x; q), wherei ≥ 2 is even. We have 0 ≤ f1 (λ) ≤ i − 1. For each v with 1 ≤ v ≤ i/2, if f1 (λ) = 2v − 1(k,i)(k,i)
or 2v − 2 then f1 (λ) = 2v − 1. In the case v = i/2, we have f2 (λ) ≤ k − i and f1 (µ) iseven. Removing the 1’s and subtracting one from each remaining part we see that these pairs
are generated by ((xq)i−2 + (xq)i−1 )/(1 − ax2 q 2 )Sbk,k−i+1 (a; xq; q). Now for 1 ≤ v ≤ (i − 2)/2,(k,i)
we have f1 (µ) = 0 and f2 (λ) ≤ k − 2v. Again removing the 1’s and subtracting one fromeach part, these pairs are generated by (xq)2v−2 (1 + xq)Sbk,k−2v+1 (a; xq; q). This gives (2.4).
To prove (2.5), suppose that (λ, µ) is a partition pair counted by Sbk,i (a; x; q), where i ≥ 3(k,i)
is odd. For each v with 1 ≤ v ≤ (i − 1)/2, if f1 (λ) = 2v or 2v − 1, then f1 (λ) = 2v. The
argument now proceeds as above, except that we have left out the case f1 (λ) = 0 beacuse i
is odd. This accounts for the extra term Sbk,k (a; xq; q). This concludes the proof of Theorem
1.3.
We now turn to Corollary 1.5. First, setting a = 1/q and x = 1 on the right-hand side of(1.1), we have
1 X (k−1)n2 +(k−i)nq(1 + q (2n+1)(i−1) )Sk,i (1/q; 1; q) =(q)∞n≥0===1 X (k−1)n2 +(k−i)nq(q)∞n∈Zi−1(−q , −q 2k−i+1 , q 2k−2 ; q 2k−2 )∞X(q)∞nAk,i (n)q ,n≥0where the penultimate line follows from the Jacobi triple product identity (1.6). On the other
hand, if we let a = 1/q and x = 1 in Theorem 1.3 and consider the effect on partition pairs
counted by sk,i (t, m, n), then parts j +1 occurring an even number of times in µ may be regarded
as repeatable parts of the form 2j + 1, while the eventual leftover occurrence of j + 1 becomes
j. This gives the pairs counted by Bk,i (n) and completes the proof of Corollary 1.5.
3. Conclusion
In addition to Andrews’ Jλ,k,i (a1 , a2 , . . . , aλ ; x; q), there are several other families of q-series
whose q-difference equations are worth exploring. We indicate three of these here. First, Andrews has developed q-difference equations for some series Kλ,k,i (a1 , a2 , . . . , aλ ; x; q) [7, Section3] which may be regarded as bilateral series analogues of the Jλ,k,i (a1 , a2 , . . . , aλ ; x; q). Second, Bressoud’s Fλ,k,i (c1 , c2 , a1 , a2 , . . . , aλ ; x; q) [15] reduce to Andrews’ Jλ,k,i (a1 , a2 , . . . , aλ ; x; q)
when c1 , c2 → ∞ and x = xq. When c1 → ∞ and c2 = −q, q-difference equations and their
combinatorial implications have been worked out for λ = 1 in [17] and for λ = 2 in [24]. Surely

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JEREMY LOVEJOY

many more instances of Bressoud’s series satisfy meaningful q-difference equations. Finally,
there are nice q-difference equations for a family of series containing both F1,k,i (−q, ∞, a1 ; xq; q)
and F1,k,i (∞, ∞, a1 ; xq; q) presented in [17, Section 6].
Acknowledgement
The author is indebted to Ae Ja Yee for her reading of an earlier version of this paper.
References[1] G.E. Andrews, An analytic proof of the Rogers-Ramanujan-Gordon identities, Amer. J. Math. 88 (1966),844-846.
[2] G.E. Andrews, Some new partition theorems, J. Combin. Theory 2 (1967), 431-436.
[3] G.E. Andrews, Partition theorems related to the Rogers-Ramanujan identitites, J. Combin. Theory 2 (1967),422-430.
[4] G.E. Andrews, A generalization of the Göllnitz-Gordon partition identities, Proc. Amer. Math. Soc. 8 (1967),
945–952.
[5] G.E. Andrews, On q-difference equations for certain well-poised basic hypergeometric series, Quart. J. Math.
19 (1968), 433-447.
[6] G.E. Andrews, A generalization of the classical partition theorems, Trans. Amer. Math. Soc. 145 (1968),
205-221.
[7] G.E. Andrews, Applications of basic hyoergeometric series, SIAM Review 16 (1974), 441-484.
[8] G.E. Andrews, On the general Rogers-Ramanujan theorem, Mem. Amer. Math. Soc. 152 (1974), 86 pp.
[9] G.E. Andrews, On the Alder polynomials and a new generalization of the Rogers-Ramanujan identities,
Trans. Amer. Math. Soc. 204 (1975), 40-64.
[10] G.E. Andrews, The hard-hexagon model and the Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci.
78 (1981), 5290-5292.
[11] G.E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), 267-283.
[12] G.E. Andrews, Parity in partition identities, Ramanujan J., to appear.
[13] G.E. Andrews and J.P.O. Santos, Rogers-Ramanujan type identities for partitions with attached odd parts,Ramanujan J. 1 (1997), 91-99.
[14] D.M. Bressoud, A generalization of the Rogers-Ramanujan identities for all moduli, J. Combin. Theory 27(1979), 64-68.
[15] D.M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer.
Math. Soc. 227 (1980), 54pp.
[16] W.G. Connor, Partition theorems related to some identities of Rogers and Watson, Trans. Amer. Math. Soc.
214 (1975), 95-111.
[17] S. Corteel, J. Lovejoy, and O. Mallet, An extension to overpartitions of the Rogers-Ramanujan identities foreven moduli, J. Number Theory 128 (2008), 1602-1621.
[18] S. Corteel and O. Mallet, Overpartitions, lattice paths, and the Rogers-Ramanujan identities, J. Combin.
Theory Ser. A 114 (2007), 14071437.
[19] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990.
[20] J. Lovejoy, Gordon’s theorem for overpartitions, J. Combin. Theory Ser. A 103 (2003), 393-401.
[21] J. Lovejoy, Overpartition theorems of the Rogers-Ramanujan type, J. London Math. Soc. 69 (2004), 562-574.
[22] J. Lovejoy, Overpartition pairs, Ann. Inst. Fourier (Grenoble) 56 (2006), 781-794.
[23] J. Lovejoy, Partitions and overpartitions with attached parts, Arch. Math. (Basel) 88 (2007), 316-322.
[24] J. Lovejoy and O. Mallet, Overpartition pairs and two classes of basic hypergeometric series, Adv. Math. 217(2008), 386-418.
[25] L.J. Rogers, Proof of certain identities in combinatory analysis, Proc. Cambridge Phil. Soc. 19 (1919), 211214.
[26] A. Selberg, Über einige arithmetische Identitäten, Avhl. Norske Vid. 8 (1936), 23 pp.

PARTITIONS WITH ROUNDED OCCURRENCES AND ATTACHED PARTS

CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE
E-mail address: lovejoy@liafa.jussieu.fr7