LACUNARY PARTITION FUNCTIONSJEREMY LOVEJOY
Abstract. We combine the theory of Bailey chains and the theory of binary quadratic forms
to show that there are large classes of q-series which are not theta series but whose coefficientsare almost all 0. We interpret some examples in terms of simple partition functions.

1. introduction
A partition of n into distinct parts is a decreasing sequence of natural numbers whose sum
is n. The rank of such a partition is the largest part minus the number of parts. Let E(n)(resp. O(n)) denote the number of partitions of n into an even (resp. odd) number of distinct
parts. It is a celebrated fact in the theory of partitions thatn = k(6k + 1)
1,E(n) − O(n) = −1, n = (2k + 1)(3k + 2)0,otherwise
This can be proven combinatorially [10] and is equivalent to Euler’s pentagonal numbertheorem,∞∞YX(1 − q n ) =(−1)n q n(3n+1)/2 .
(1.1)
n=−∞

n=1

In particular, we have E(n) − O(n) = 0 for almost all natural numbers. There are, in fact,
numerous examples like (1.1) of simple partition functions given by theta functions or false
theta functions (see [3], for instance) which are trivially almost always 0. A function on N
which is almost always 0 is called lacunary, and the modern study of lacunary q-series has
its origins in the work of Ramanujan on eta functions.
If one asks for lacunary q-series and partition functions which are not simply theta or
false theta series, then one can identify several examples by using the theory of complex
multiplication [14]. For instance, Serre [15] and Gordon and Robins [12] have continued thework of Ramanujan and identified approximately 60 pairs (r, s) for which∞Y(1 − q n )r (1 − q 2n )sn=1

is lacunary. Unfortunately, the associated partition functions can be rather unnatural. (Seealso [16, 17].)
It turns out that lacunary partition functions can also arise from q-series with “real multiplication.” In a more combinatorially relevant example, Andrews, Dyson, and Hickerson [6]Date: April 1, 2021.
1

2JEREMY LOVEJOY√used the arithmetic of Q( 6) to show that the number of partitions into distinct parts witheven rank minus the number of partitions into distinct parts with odd rank is almost always
0 (and, in fact, is equal to any given integer infinitely often). (See also [9].)Here we shall observe how a theorem on quadratic forms can be combined with the theoryof Bailey chains [4] to give a general technique for producing families of lacunary q-series
which correspond to natural and reasonably simple partition functions. Theorems 1 and 2
below are just two such examples. We employ the standard q-series notation,(a1 , ..., aj ; q)n =n−1Y(1 − a1 q k )...(1 − aj q k ).

k=0

Theorem 1. For k ≥ 1 we have
∞ XnX

2

2(−1)n+j q kn +(k−1)n+j (1 − q 2n+1 )n=0 j=−n= 2Xnk ≥...≥n1 ≥0(q; q)nk (−1)nk q nk (nk +1)/2+nk−1 (nk−1 +1)+...+n1 (n1 +1).
(q; q)nk −nk−1 ...(q; q)n2 −n1 (q; q)n1 (1 + q n1 )Theorem 2. For k ≥ 1 we have
∞ XnX

2(−1)n+j q ((2k+3)n +(2k+1)n)/2−j(3j+1)/2 (1 − q 2n+1 )n=0 j=−n=

Xnk ≥...≥n1 ≥0(q; q)nk (−1)nk q nk (nk +1)/2+nk−1 (nk−1 +1)+...+n1 (n1 +1).
(q; q)nk −nk−1 ...(q; q)n2 −n1

The lacunarity of each series in Theorem 1 and almost every series in Theorem 2 will follow
immediately fromTheorem 3. Take a set S ⊂ Z × Z and a function ω : Z × Z → Z. IfXX22a(n)q n =ω(x, y)q ax +bx+cy +dy(1.2)(x,y)∈S

converges in some disk around q = 0 and −ac is not a square, then|{n ≤ N : a(n) 6= 0}|  √N.
log N

The partition theoretic interpretation of the generating functions in Theorems 1 and 2 forarbitrary k is detailed in §3 and relies on the Durfee dissection of a partition [2]. The casek = 1 of Theorem 1 is a classical result which relates the number of solutions to x2 + y 2 = n
to the number of ways to write the natural number n as the sum of consecutive positive
integers with even largest summand minus the number of ways to write to write n as the sumof consecutive positive integers with odd largest summand.
To interpret the generating function for the k = 1 case of Theorem 2, let A(n) denote thenumber of partitions of n into distinct parts which differ by at most 2 and whose smallest
part is at most 2. The difference condition is the opposite of that occurring in the famous

LACUNARY PARTITION FUNCTIONS

3Rogers-Ramanujan identities. These partitions are enumerated by one of Ramanujan’s fifth
order mock theta functions [5],
ψ1 (q) =∞X

q n(n+1)/2 (−q; q)n ,n=0

and the right hand side of Theorem 2 for k = 1 is then the generating function for the numberof partitions counted by A(n) with largest part even minus the number of partitions countedby A(n) with largest part odd.
If one investigates the values of the relevant partition functions, it quickly becomes apparent
that there is little hope of finding a combinatorial explanation for their lacunarity. This
property arises instead from estimates involving norm functions in quadratic fields which
imply Theorem 3. These estimates and Bailey chains are considered in the following section
before the main theorems are established in §3.
2. Quadratic forms and Bailey chains
The following theorem seems to have first appeared in print in [13] and is credited there
to one P. Bernays [8].
Theorem 4. Let Q(x, y) = Ax2 + Bxy + Cy 2 be a quadratic form which is not negativedefinite and whose discriminant is not a square. If P (N ) denotes the number of positive
integers less than N which are integrally representable by Q, thenP (N )  √N.
log N

Proof of Theorem 3. Multiplying by appropriate powers of q and making substitutions of
the form q → q ` in (1.2) allows one to complete the square and apply the above theorem. In order to use Theorem 3 to establish the lacunarity of q-series and partition functions,
we need to produce representations for q-series which resemble counting functions for binary
quadratic forms. Below we give details about one general method by which this can beaccomplished.
Two sequences (αn , βn ) form a Bailey pair with respect to a if the following two equivalentconditions hold:nXαrβn =,(2.1)(q; q)n−r (aq; q)n+r
r=0n

αn =

(1 − aq 2n )(a; q)n (−1)n q n(n−1)/2 X −n(q ; q)j (aq n ; q)j q j βj .
(1 − a)(q; q)n(2.2)

j=0

The condition (2.1) is the original definition of a Bailey pair, while the equivalent form
(2.2) is demonstrated in [1].
Proposition 5. If (αn , βn ) form a Bailey pair with respect to a, then so do0

αn =(b1 , c1 ; q)n (aq/b1 c1 )n αn(aq/b1 , aq/c1 ; q)n(2.3)

4

JEREMY LOVEJOY

andnX (b1 , c1 ; q)j (aq/b1 c1 ; q)n−j (aq/b1 c1 )j βj1βn =.
(aq/b1 , aq/c1 ; q)n(q; q)n−j0(2.4)

j=0Since each Bailey pair leads to a new Bailey pair, we can indefinitely iterate to obtain theso-called Bailey chain.
Theorem 6 (Andrews, [4]). If (αn , βn ) form a Bailey pair with respect to a, thenaq
X (b1 , c1 , ..., bk , ck , q −n ; q)r  −ak q k+n r( aqbk , ck ; q)nq r(r−1)/2 αraq
aq aqn+1 ; q)
,,...
,,aq
(aq, bkaqck ; q)n( aqbc...bcr
1 1k kb1 c1
bk ckr≥0=×X(q −n ; q)nk (bk , ck ; q)nk ...(b1 , c1 ; q)n1aq
bk ck q −naqaq)nk ( bk−1, ck−1; q)nk ...( aqnk ≥nk−1 ≥...≥n1 ≥0 (ab1 , c1 ; q)n2
nk−1 ; q)n2 −n1  aq
( bk−1aqck−1 ; q)nk −nk−1 ...( baqaq n1
1 c1(q; q)nk −nk−1 ...(q; q)n2 −n1

bk−1 ck−1...

b1 c1

q nk βn1 .

All general statements about q-series such as Theorem 6 are subject to convergence conditions, though we shall not state them explicitly.
Theorem 7. If
βn =(d, a/bc; q)n(q, dq, a/b, a/c; q)n

(2.5)

andn

(a/d; q)n (1 − aq 2n )(−d)n q n(n−1)/2 X (a; q)j−1 (b, c, d; q)j (1 − aq 2j−1 )aj,αn =(1 − a)(dq; q)n(q, a/b, a/c, a/d; q)j (bcd)j(2.6)

j=0then (αn , βn ) form a Bailey pair with respect to a.
Proof. In Watson’s q-analogue of Whipple’s transformation [11, p.242, Eq. (III.18)], let
a = a/q and then let e = aq n . The theorem then follows from the second definition of aBailey pair (2.2).
Theorem 8. Ifβn =(bcdq/a; q)n(bq, cq, dq; q)n(2.7)

andnαn =(a/b, a/c, a/d; q)n (1 − aq 2n )(−bcdq)n q n(n−1)/2 X (a; q)j−1 (b, c, d; q)j (1 − aq 2j−1 )aj,(1 − a)(bq, cq, dq; q)n an(q, a/b, a/c, a/d; q)j (bcd)jj=0(2.8)
then (αn , βn ) form a Bailey pair with respect to a.
Proof. Apply Proposition 5 to the Bailey pair from the previous theorem withP b1 = a/b andc1 = a/c. Then (2.8) is straightforward. To get (2.7), notice that if [z n ] a(j)z j := a(n),LACUNARY PARTITION FUNCTIONS5then we can write (2.4) as0βn =X (d, a/bc; q)(bcqz/a)j X (bcq/a; q)j z j1[z n ](bq, cq; q)n(q, dq; q)j(q; q)jj

= [z n ]=j(bcqz/a; q)∞(bq, cq; q)n (z; q)∞X (d, a/bc; q)(bcqz/a)j(q, dq; q)jj

X (bcdq/a; q)j z j1[z n ](bq, cq; q)n(dq; q)j(by [11, p. 236 Eq. (II.3)])(by [11, p. 241 Eq. (III.3)]).

jTheorems 7 and 8 can now be employed to give numerous Bailey pairs which are related
to quadratic forms. The case d = 0 has been useful in establishing relationships between
Ramanujan’s fifth, sixth, and seventh order mock theta functions and indefinite quadraticforms, and several examples of Bailey pairs corresponding to indefinite forms are given in[5, 7]. By letting some of the variables tend to ∞ we instead find Bailey pairs with positive
definite quadratic forms. The following lemmas give two examples, the positive definite form
used to prove Theorem 1 and the indefinite form which leads to Theorem 2.
Lemma 9. Ifβn =2(q; q)n (1 + q n )andn(1 − q 2n+1 )q n(n−1)/2 X
2(−1)j q j ,αn =(1 − q)j=−nthen (αn , βn ) is a Bailey pair with respect to q.
Proof. Let d = −1, a = q, and b, c → ∞ in Theorem 7.

Lemma 10. If
βn = 1and2nq 2n +n (1 − q 2n+1 ) X
αn =(−1)j q −j(3j+1)/2 ,(1 − q)j=−nthen (αn , βn ) is a Bailey pair with respect to q.
Proof. Let a = q and d = 0 in Theorem 8, and then let b, c → 0. (See also [5, p.121,(5.11)].)
3. Durfee dissection and proofs of the main theorems
Proofs of Theorems 1 and 2. Simply insert the Bailey pairs from Lemmas 9 and 10 into
the Bailey chain (Theorem 6), let a, bk = q, and let ck , bk−1 , ck−1 , ..., b1 , c1 , n → ∞.
To establish the partition-theoretic interpretation of Theorems 1 and 2, we require theDurfee rectangle dissection of a partition [2]. In the Ferrers diagram of a partition λ, wecall the largest n × (n + 1) rectangle (upper left justified) the Durfee rectangle. To obtain

6JEREMY LOVEJOY

Figure 1. A Durfee rectangle dissection
the Durfee rectangle dissection of λ, notice that to the right of the Durfee rectangle will be
another partition, and we call its Durfee rectangle the second Durfee rectangle of λ. Clearly
we can continue until we run out of rectangles.
Definition 11. A partition λ is called k-admissible provided(i) If there are less than k Durfee rectangles, then there is nothing to the right of the finalrectangle.
(ii) For 1 ≤ j ≤ k, there is nothing directly below the lower right dot of the jth Durfee
rectangle.
The Durfee rectangle dissection of a partition which is 1−, 2−, and 3−admissible, but notk−admissible for k ≥ 4, is shown in Figure 1. We should point out that to facilitate ourdiscussion the definition of admissibility differs slightly from that in [2].
For any partition λ into j distinct parts, let the associated partition be that which isobtained by subtracting i from the ith smallest part for 1 ≤ i ≤ j. Let the k-largest partof λ denote the number of parts plus the number of columns to the right of the kth Durfee
rectangle in the associated partition. The 0-largest part is just the largest part.
We begin with the partitions enumerated in Theorem 2. Let Ak (n) denote the numberof partitions λ of n into distinct parts whose associated partition is k-admissible and suchthat the columns to the right of the kth Durfee rectangle of the associated partition form a−partition into distinct parts. Let A+k (n) (resp. Ak (n)) be the number of partitions countedby Ak (n) with even (resp. odd) k-largest part.
Theorem 12. When 3(2k + 5) is not a square, we have−|{n ≤ N : A+k (n) 6= Ak (n)}|  √N.
log N

Proof. From Andrews [2], we know that the generating function for k-admissible partitions
with at most N parts and nothing to the right of the final rectangle isXN ≥nk ≥nk−1 ≥...n1 ≥0(q; q)N q nk (nk +1)+...+n1 (n1 +1)(q; q)N −nk (q; q)nk −nk−1 ...(q; q)n2 −n1 (q; q)n1(3.1)

LACUNARY PARTITION FUNCTIONS7To make a partition into N distinct parts with a k-admissible associated partition and track
the parity of the number of parts, we multiply by (−1)N q N (N +1)/2 . Then we multiply by(q; q)n1 to ensure that the columns to the right of the kth Durfee rectangle form a partition
into distinct parts and to track the parity of the number of such columns. If we sum over all
N , then we find that the k-fold summation on the right side of Theorem 2 is the generating−function for A+k−1 (n) − Ak−1 (n). To complete the proof, invoke Theorem 3.
A similar argument reveals the nature of the partitions generated by the q-series in Theorem
1. Let Rk (n) denote the number of partitions λ of n into distinct parts whose associatedpartition is k-admissible with less than k Durfee rectangles plus twice the number of partitions
λ of n into distinct parts whose associated partition is k-admissible with exactly k Durfee
rectangles such that the first n1 parts of λ are n1 consecutive integers, where n1 is the size ofthe kth Durfee rectangle in the associated partition. Let Rk+ (n) (resp. Rk− (n)) be the numberof partitions counted by R(n) with even (resp. odd) k-largest part.
Theorem 13. We have|{n ≤ N : Rk+ (n) 6= Rk− (n)}|  √N.
log N

Proof. The argument is nearly the same as in Theorem 12, a notable difference being that thepositive-definite forms require no restrictions on values of k to guarantee lacunarity. After
multiplying (3.1) by (−1)N q N (N +1)/2 , we multiply by by 2/(1 + q n1 ), which is 1 if there aren’tk Durfee rectangles. If there are k Durfee rectangles this ensures that the first n1 parts areconsecutive integers, tracks the parity of the number of columns to the right of the final
rectangle in the associated partition, and counts the partition twice.
References[1] G.E. Andrews, Connection coefficient problems and partitions, Proc. Symp. Pure Math. 34 (1979), 1-24.
[2] G.E. Andrews, Partitions and Durfee dissection, Amer. J. Math. 101 (1979), 735-742.
[3] G.E. Andrews, Partitions: yesterday and today, New Zealand Math. Soc., Wellington, 1979.
[4] G.E. Andrews, q-series: Their development and application in analysis, number theory, combinatorics,physics, and computer algebra, CBMS 66, 1984.
[5] G.E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc. 293 (1986),
113-134.
[6] G.E. Andrews, F.J. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math.
91 (1988), 391-407.
[7] G.E. Andrews and D. Hickerson, Ramanujan’s lost notebook VII: the sixth order mock theta functions,Adv. Math. 89 (1991), 60-105.
[8] P. Bernays, Über die Darstellung ... durch primitiven binären quadratischen Formen, Dissertation,Göttingen, 1912.
[9] H. Cohen, q-identities for Maass waveforms, Invent. Math. 91 (1988), 409-422.
[10] F. Franklin, Sur le développement du produit infini (1 − x)(1 − x2 )(1 − x3 )..., Comptes Rendus 82 (1881),
448-450.
[11] G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990.
[12] B. Gordon and S. Robins, Lacunarity of Dedekind η-products, Glasgow Math. J. 37 (1995), 1-14.
[13] R.W.K. Odoni, On norms of integers in a full module of an algebraic number field and the distributionof values of binary integral quadratic forms, Mathematika 22 (1975), 108-111.
[14] K. Ribet, Galois representations attached to eigenformas with Nebentypus, Lect. Notes Math. 601 (1977),
17-52.
[15] J.-P. Serre, Sur la lacunarité des puissances de η, Glasgow Math. J. 27 (1985), 203-221.

8JEREMY LOVEJOY[16] M. Tamba, On a partition function which assumes all integral values, J. Number Th. 41 (1992), 77-86.
[17] M. Tamba, Note on a partition function, J. Number Th. 41 (1992), 280-282.
Projet Theorie des Nombres, Institut de Mathematiques de Jussieu, Case 247, 4 Place Jussieu,75252 Paris CEDEX 05
Email address: lovejoy@math.jussieu.fr