PARTITIONS WITH DIFFERENCE CONDITIONS AND ALDER’S
CONJECTURE
AE JA YEE
Abstract. In 1956, Alder conjectured that the number of partitions of n into parts differing by at
least d is greater than or equal to that of partitions of n into parts ≡ ±1 (mod d + 3) for d ≥ 4.
In 1971, Andrews proved that the conjecture holds for d = 2r − 1, r ≥ 4. We sketch a proof of the
conjecture for d ≥ 32.
1. Introduction
The well-known Rogers-Ramanujan identities may be stated partition-theoretically
as follows. If c = 1 or 2, then the number of partitions of n into parts ≡ ±c (mod 5)
equals the number of partitions of n into parts ≥ c with minimal difference 2 between
parts. In 1926, I. Schur proved that the number of partitions of n with minimal
difference 2 between parts and no consecutive multiples of 3 equals the number of
partitions into parts ≡ ±1 (mod 6).
In 1956 [2], H. L. Alder posed the following problems. Let qd (n) be the number of
partitions of n into parts differing by at least d; let Qd (n) be the number of partitions
of n into parts ≡ ±1 (mod d + 3); let ∆d (n) = qd (n) − Qd (n). (a) Is ∆d (n) nonnegative
for all positive d and n? It is known that ∆1 (n) = 0 for n > 0 by Euler’s identity, that
the number of partitions of n into distinct parts equals that of partitions of n into odd
parts, ∆2 (n) = 0 for n > 0 by the first Rogers-Ramanujan identity, and ∆3 (n) ≥ 0
for all n > 0 by Schur’s theorem, which states that ∆3 (n) equals the number of those
partitions of n into parts differing by at least 3 which contain at least one pair of
consecutive multiples of 3. (b) If (a) is true, can ∆d (n) be characterized as the number
of partitions of n of a certain type, as is the case for d = 3?
In 1971 [4], G. E. Andrews gave some partial answers to problem (a).
Theorem 1.1 (Andrews [4]). For any d ≥ 4, lim ∆d (n) = ∞.
n→∞
Theorem 1.2 (Andrews [4]). If d = 2r − 1, r ≥ 4, then ∆d (n) ≥ 0 for all n.
To prove Theorem 1.2, Andrews studied the set of partitions of n into distinct parts
≡ 2i (mod d) for 0 ≤ i < r, the size of which is greater than or equal to Qd (n) for
any n, and succeeded in finding a set of partitions with difference conditions between
parts that are complicated but much stronger than the difference condition for qd (n).
He then proved that partitions into distinct parts ≡ 2i (mod d) for 0 ≤ i < r and
partitions with the difference conditions are equinumerous. As a result, he was able to
prove Alder’s conjecture in these particular cases.
By generalizing Andrews’ methods and constructing an injection, we are able to
prove that Alder’s conjecture holds for d = 7 and d ≥ 32 (see [8]). Therefore, Alder’s
conjecture still remains unresolved for 4 ≤ d ≤ 30 and d 6= 7, 15.
In section 2, we will give an outline of the proof of the case when d = 2r − 1, r ≥ 4
by Andrews, and in Section 3 we will sketch the proof of the case when d ≥ 32.
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2
AE JA YEE
In the sequel, we assume that |q| < 1 and use the customary notation for q-series
(a)0 := (a; q)0 := 1,
∞
Y
(a)∞ := (a; q)∞ :=
(1 − aq k ),
k=0
(a)n := (a; q)n :=
(a; q)∞
(aq n ; q)∞
for any n.
2. The case when d = 2r − 1
From the definitions of qd (n) and Qd (n), the generating functions for qd (n) and Qd (n)
are, respectively,
∞
X
n
qd (n)q =
n=0
∞
X
Qd (n)q n =
n=0
n
∞
X
q d( 2 )+n
n=0
(q)n
,
1
(q; q d+3 )∞ (q d+2 ; q d+3 )∞
(see [5, Chap.1].)
For a given d = 2r − 1, define
∞
X
r−1
Ld (n)q n = (−q; q d )∞ (−q 2 ; q d )∞ · · · (−q 2 ; q d )∞ .
fd (q) =
n=0
Then
fd (q) =
1
(q; q 2d )∞ (q d+2 ; q 2d )∞ (q d+4 ; q 2d )∞
· · · (q d+2r−1 ; q 2d )∞
.
Let
Ad = {m | m ≡ 2i
(mod d),
0 ≤ i ≤ r − 1}
and
A0d = {m | m ≡ i
(mod d),
1 ≤ i ≤ 2r − 1}.
Let βd (m) denote the least positive residue of m modulo d. For m ∈ A0d , let b(m) be
the number of terms appearing in the binary representation of m, and let v(m) denote
the least 2i in this representation. We need a theorem of Andrews [3], which we state
without proof in the following theorem.
Theorem 2.1. Let D(Ad ; n) denote the number of partitions of n into distinct parts
taken from Ad , and let E(A0d ; n) denote the number of partitions of n into parts taken
from A0d of the form n = λ1 + λ2 + · · · + λs , such that
λi − λi+1 ≥ d · b(βd (λi+1 )) + v(βd (λi+1 )) − βd (λi+1 ).
Then D(Ad ; n) = E(A0d ; n).
(2.1)
ALDER’S CONJECTURE
3
Meanwhile, Andrews [4] established the following general theorem in the theory of
partitions.
∞
Theorem 2.2. Let S = {ai }∞
i=1 and T = {bi }i=1 be two strictly increasing sequences of
positive integers such that b1 = 1 and ai ≥ bi for all i. Let ρ(S; n) and ρ(T ; n) denote
the numbers of partitions of n into parts taken from S and T , respectively. Then, for
n ≥ 1,
ρ(T ; n) ≥ ρ(S; n).
By comparing parts arising in partitions counted by Qd (n) and Ld (n), we see that
Ld (n) ≥ Qd (n) for r ≥ 4. Since Ld (n) = D(Ad ; n) and qd (n) ≥ E(A0d ; n), it follows
from Theorem 2.1 that qd (n) ≥ Ld (n). Therefore, we have shown that qd (n) ≥ Qd (n)
for d = 2r − 1, r ≥ 4.
3. The case when d 6= 2r − 1
We denote the coefficient of q n in an infinite series s(q) by [q n ](s(q)).
For a given d, uniquely define the integer r by
2r < d + 1 < 2r+1 ,
and let L0d (n) be the number of partitions of n into distinct parts ≡ 1, 2, 4, . . . , 2r−1
(mod d). Then the generating function fd (q) for L0d (n) is
∞
X
r−1
L0d (n)q n = (−q, −q 2 , −q 4 , . . . , −q 2 ; q d )∞ .
fd (q) =
n=0
By examining the generating function fd (q), we find from Theorem 2.2 that for
d ≥ 32,
L0d (n) + L0d (n − 2r ) ≥ Qd (n),
n ≥ 1,
where L0d (m) = 0 if m ≤ 0. Thus, we only need to prove that
qd (n) ≥ L0d (n) + L0d (n − 2r ).
(3.1)
Let X(d; n) and Y (d; n) be the sets of partitions of n counted by qd (n) and L0d (n),
respectively. Then
X(d; n) ⊃ Y (d; n),
since the conditions for Y (d; n) are much stronger than those for X(d; n). Thus, if there
exists an injection from Y (d; n − 2r ) to X(d; n) \ Y (d; n), then the Alder conjecture
holds.
Lemma 3.1. For any d ≥ 32 not of the form 2r − 1,
qd (n) ≥ L0d (n) + L0d (n − 2r ),
n ≥ 4d + 2r .
To obtain Lemma 3.1, we construct an injection from Y (d; n − 2r ) to X(d; n) such
that the image of a partition in Y (d; n − 2r ) does not satisfy the conditions for Y (d; n).
For n < 4d + 2r , we can show that qd (n) ≥ Qd (n) by merely counting qd (n) and
Qd (n). Therefore, from Lemma 3.1, inequality (3.1), and Andrews’ result we obtain
the following theorem.
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AE JA YEE
Theorem 3.2. For d ≥ 31,
qd (n) ≥ Qd (n),
n ≥ 1.
4. Conclusion
The most intriguing identities in the theory of partitions have been the RogersRamanujan identities. Extremely motivated by these identities, Schur searched for
further analogous partition identities. In 1926 [7], Schur proved that the number of
partitions of n with minimal difference 3 between parts and no consecutive multiples
of 3 equals the number of partitions of n into parts ≡ ±1 (mod 6).
Alder’s conjecture has naturally arisen from the Rogers-Ramanujan identities and
the Schur identity, and has been open for about 50 years. The difficulty in dealing
with the conjecture is that no set S exists such that the number of partitions of n with
parts from S is equal to qd (n) for d ≥ 3 (see [1, 6].) Thus finding injections between
two sets might not be the most efficient method, but it is the best approach for the
conjecture at this point.
Acknowledgment. The author thanks George E. Andrews for suggesting that she
work on this project and Bruce C. Berndt for his comments and advice.
References
[1] Alder, H. L. (1948) The nonexistence of certain identities in the theory of partitions and compositions, Bull. Amer. Math. Soc., 54, 712–722.
[2] Alder, H. L. (1956) Research problem no. 4, Bull. Amer. Math. Soc., 62, 76.
[3] Andrews, G. E. (1969) A general theorem on partitions with difference conditions,
Amer. J. Math., 91, 18–24.
[4] Andrews, G. E. (1971) On a partition problem of H. L. Alder, Pacific. J. Math., 36, 279–284.
[5] Andrews, G. E. (1998)The Theory of Partitions, Addison–Wesley, Reading, MA, 1976; reissued:
Cambridge University Press, Cambridge.
[6] Lehmer, D. H. (1946) Two nonexistence theorems on partitions, Bull. Amer. Math. Soc., 52,
538–544.
[7] Schur, I. (1917) Ein Beitrag zur additiven Zahlentheorie, S.-B. Akad. Wiss. , Berlin, 301–321.
[8] Yee, A. J. On Alder’s conjecture, submitted for publication.
Department of Mathematics, The Pennsylvania State University, University Park,
PA 16802, USA
E-mail address: [email protected]