J OURNAL DE T HÉORIE DES N OMBRES DE B ORDEAUX

J.-L. N ICOLAS
A. S ÁRKÖZYOn partitions without small parts
Journal de Théorie des Nombres de Bordeaux, tome 12, no 1 (2000),
p. 227-254.
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227-On partitions without small parts
par J.-L. NICOLAS et A. SÁRKÖZYRÉSUMÉ. On désigne par r(n, m) le nombre de partitions del’entier n en parts supérieures ou égales à m. En appliquantla méthode du point selle à la série génératrice, nous donnons
une estimation asymptotique de r (n, m) valable pour n ~ ~, et
1 ~m~c1n (logn)c2.

ABSTRACT. Let r(n, m) denote the number of partitions of n into

parts, each of which is at least m. By applying the saddle point
method to the generating series, an asymptotic estimate is given
for r(n, m), which holds for n ~ ~, and 1 ~ m ~ c1n (log n)c2

1. INTRODUCTION

Let r(n, m) (resp. p(n, m)) denote the number of partitions of the positive
integer n into parts, each of which is at least m (resp. at most m), that is,
the number of partitions of n of the typeand let q(m, n) denote the number of partitions of n into distinct parts,
each of which is at least m, i.e., the number of solutions of

The function p(n, m) has been extensively studied by Szekeres in [6] and[7].
In the last several years, Dixmier and Nicolas [1], [2] have studied thefunction r(n, m) while Erd6s, Nicolas and Szalay [4], [3] have estimated thefunction q(n, m); in all these papers m is relatively small in terms of n. In[5], Freiman and Pitman gave an asymptotic formula for q(n, m) in termsof a certain parameter a = a(n, m) in a quite wide range for m:
Manuscrit reçu le 6 mai 1999.

Partially supported by Hungarian National Foundation for Scientific Research, Grant No. T029759, by CNRS, Institut Girard Desargues, UMR 5028 and Balaton, 98009. This paper waswritten while the second author was visiting the Université Claude Bernard Lyon 1.

228

Theorem A (G.A. Freiman and J. Pitman, ~5~). As n - +oo we havewhere o- and B are given byandanduniformly with respect to m such that

Here Ko and the implied constants in the estimate of E are effective positive
constants independent of m and n.

They used a probabilistic approach in proving this theorem, however,their proof could be presented as well in terms of the saddle point method,
without any reference to probability theory. Moreover, in certain intervals
for m, they succeeded in making (1.1) more explicit by eliminating theparameter a, i.e., by giving an asymptotics for q(n, m) in terms of m andnonly.

In this paper our goal is to prove the r {n, m) analog of Theorem A and,in fact, we will prove a sharper result with a series expansion instead of afactor of type 1-~- E as in ( 1.1 ) . Before formulating our main theorem, first
we have to introduce a parameter a (playing the same role as the parametera in Theorem A).
We shall write e"’° = e(a) and N, R and C will denote the set of thepositive integers, real numbers and complex numbers, respectively. For alla E R and 0R1, writing z R e(a) clearly we have=229

where r* (~, m) n.~. Substituting a=for 01n and 0r*(t,m) :5
we obtain
0 and dividing bynfor-say. Clearly,and U(R) is continuous in (0,1). Thus U(R) has a minimum in (0,1).
Moreover, U(R) is twice differentiable in (0,1), thus its minimum is attained
at a point Ro satisfying the equationor, in equivalent form,Setting(1.4) can be rewritten in the form

Clearly we haveis decreasing in (1, +oo). Thus (1.4) determines R ( 1) uniquely.
andDefine a by--

230(with the R defined by (1.4)) so that cr &#x3E; 0, and (1.5) can be rewritten in
the form

For hEN, l E N write

so thatby (1.6),

We shall also write

Clearly, we havesay. By estimating the integral J we will prove:

Theorem 1. There exist real numbers d(x, y) with x E N, y E {1,2,...,x}such that writing

for any fixed k E N, k &#x3E; 3 as n -~ +oo u~e have

with

231

whereand, for a certain absolute constant H &#x3E; 2,uniformly with respect to m such thatIn particular, 2ve haveand for k=6 (1.12) holds with

where L2, ..., L6 are defined above and where232

In Theorem 1 and throughout the rest of the paper, the constants cl, C2,
and the constants implied by the notations O(...), «, ~ as well are effectively computable, and they may depend only on the parameter l~ in Theorem 1 but are independent of any other parameters, in particular, of m andn. (We write f « g if f - O(g), and f ~ g means that both f « g and...

g « f hold.)

Note that an upper bound will be given for the numbers A(h, l) with1 .~h and thus also for Lh in Lemma 2, and the order of magnitude ofBL12

will be determined in Lemma 3.
2
The rest of the paper will be devoted to the proof of Theorem 1, and inPart II of this paper the result will be made more explicit by eliminating
the parameter car in a wide range for m and giving a sharp estimate for thenumbers L~ in Theorem 1.
In [2], it has been proved that there exists a function 9 such that, forA &#x3E; 0, the following asymptotic estimation holds :=Moreover, the function X e g(A) - a log A was proved to be analytic in aneighbourhood of 0, so that it can be written
and, for j &#x3E; 2, the values of the coefficients aj are rational fractions of-r~’

The following relation was also given in [2] :where, in the bracket, the coefficient a2 shows up. In Part II, it will be
proved that

wheren 1/2-é, 9=

can beIn the above relation, whenever mof its power series expansion,
firstterms
replaced by the233

for instance :In Part II, the error term in the asymptotic estimation (1.19) of r(n,m)
will also be precised.
2. ESTIMATE OF Q

We shall prove

Lemma 1. Let k &#x3E; 1 be an integer. There is a positive number H
such that H &#x3E; 2 andand

Proof of Lemmas 1. For large n we havesince otherwise by (1.15) for large m we should have

which contradicts (1.6).
On the other hand, we havesince otherwise we had

again in contradiction with (1.6).
We split the sum on the left hand side of (1.6) into two parts:

=H(k)

234

say. Clearly we haveandMoreover, writing M=we havesay. The function xe-x is decreasing for x &#x3E; 1 and thus the terms in thesum
decrease as j increases. Thus we have

E’ 2Since

thus it follows from M &#x3E; 1/c, (2.3), (2.4) and (2.9) thatandThe function(2.3) we have=xe-ax is decreasing for x &#x3E; 1/Q and by (1.15) andand thus

By (2.11) and (2.12) we have

It follows from (2.8), (2.10) and (2.13) thatNow we have to distinguish two cases.

235

Case 1. Assume first that

Then by (2.5), (2.7), (2.14) and (2.15) we havewhenceIt follows from (2.15) and (2.17) that.

I-Case 2. Assume now that

Then by (2.5), (2.6), (2.14) and (2.19) we have

Writing(2.20) can be rewritten in the formBy (2.19) here we have ma &#x3E; 1, and f (x) &#x3E; 1 for x &#x3E; 1.
impliesThus (2.22)-whenceMoreover, for z - +oo (2.21) implies
and thus it follows from y - +oo, f (x) ~ y (note that this implies x -7+cxJ) that236

Thus it follows from (2.22) thatNow we are ready to prove (2.1) and (2.2). By (2.23), there is a positive
Knl/2 implies 1/Q &#x3E; m and then by (2.17), (1’number K such that msatisfies (2.1) (for mKn 1/2). On the other hand, if H is large enoughthenandm,
1/Q m by (2.18) and thus (2.2) holds by (2.24)mm, (2.1)
Hn1/2, then for(for m &#x3E; Hnl/2). Finally, ifm it follows from (2.22) and this completes
holds by (2.17) while forDthe proof of Lemma 1.
3. ESTIMATE OF THE SUMS

Lemma 2. For all K E N‘ there is a positive nurraber C2

for h = l, 2, ..., K and t==c2(K) such that

1, 2, ..., h (with H defined in Lemma 1).

Proof of Lemma 2. We split the sum in the definition of A(h, t) into twoparts:

say. Clearly,

while for m

Hen1/2 by (2.1) in Lemma 1 we have

Moreover, ifs

we have237

and thus

For m &#x3E; Hn 1/2 by (2.2) for all j in I:2 we haveso that(3.5) holds also in this case. By (3.5), writing mi=max(m, Hn 1/2)

we have

An easy consideration shows that uniformly forx
j we have-----and thus

so thatSince we have(which can be shown easily by induction on h) thus it follows from (3.6)
that

1. Moreover,
By Lemma 1, it follows from the definition of mi thatthat
1
followsfromLemma
have
Thus
it
weaby(3.7)0(1).
=

whence, again by Lemma 1 and the definition of m1, we have
1. /,,and238(3.1) follows from (3.2), (3.3), (3.4), (3.8) and (3.9) which completes theproof of Lemma 2.

0

4. ESTIMATE OF B

Lemma 3. We have

Proof of Lemma 3. By Lemma 1 and (1.15) we have m + 1/ u
again by Lemma 1,n and thus,

which completes the proof of the lemma.
5. ESTIMATE OF THE INTEGRAND FAR FROM 0

We split the integral J defined in (1.9) into two parts. Set

where c* is a positive number, large enough in terms of k, which will be
determined later. Letandso that

239and write

say. We will show that the integral Ji gives the main contribution while J2
gives only a small error term. Thus we shall need an asymptotic formulafor J1 and an upper bound for J2. First we will estimate the integrand inJ2.
Lemma 4. For

2 zve have

1/2, rrc E N‘,

Proof of Lemmas 4. This is Lemma 8 in [5].
Lemma 5.

Let 17 be defined by (5.1~. Uniformly for

we have

Proof of Lemma 5. For all 0
havewhencer

1 and # E IEB, writing

240

since (1 - r)2

1. It follows thatsince by (1.15) and Lemma 1 we haveFor mjm +[1/u], the following inequality holds:

and this implies

For 0x2 a simple calculation shows that

and, since,

it follows from (5.6) that

By Lemma 1, Lemma 4, (5.1) and (5.3), it follows thatby (2.1), am 0(l), and from (2.1) and&#x3E;
nsothat, by choosing c* large enough (in terms of(5.1), r~2o~-3 c3c*2 logk), (5.8) yields (5.4).

Now, in the first case (m=241

In the second case (m &#x3E; H ý1ï), by (2.2)

and from (2.2) and (5.1)

since mn, and again, by choosing c* large enough, (5.8) yields (5.4),
Dwhich completes the proof of the lemma.
6. ESTIMATE OF J2Lemma 6. We have

Proof of Lemma 6. By Lemma 5 we have

Moreover, by Lemma 2 and (1.8) we have(6.1) follows from (6.2) and (6.3).
7. ESTIMATE OF THE INTEGRAND NEAR 0
For mjn, write

242

so thatand the integrand in J can be rewritten asn

Lemma 7.

Uniformly for(7.3)
wehave(7.4)
and(7.5)
Proof of Lemma 7. For all a satisfying (7.3) we haveso thatwhich proves (7.4).
Since ex - 1 &#x3E; x for x &#x3E; 0 thus it follows from (7.4) thatwhich proves the first inequality in (7.5). Finally, q lao(l) follows from0theandthisofLemma 7.
andLemma
1,
completes
proof(1.15), (5.1)=

243

Lemma 8.

Uniformly for a satisfying (7.3) and rrt jn we have

Proof of Lemma 8. For all j we haveand thus

For j &#x3E; 1 /10q we have

The first inequality in (7.6) follows from (7.7), (7.8) and (7.9).
For i 1/cr clearly we havesince q la

j &#x3E;=0(1) as we have seen in the proof of Lemma 7. Moreover, for

we havek + 1, and thee-I is maximal
For 0 ~ the function fk(x)e-(k+1).
it follows
Thusofthevalue
maximum is fk(k + 1) = (k +from (7.11) that==1)k+l(7.10) and (7.12) complete the proof of (7.6).

244

Lemma 9. If k e N,andthen we havewith

Proof of Lenrama 9. By (7.14) we havewhereandBy (7.14) clearly we have

245

Moreover, by (7.13), (7.14) and (7.15) we have10i-11(7.16) and (7.17) follow from (7.18), (7.19) and (7.20), and this completesD
the proof of the lemma.
Lemma 10.

Uniformly for lal r~ the integrand in Ji iswhere(7.22)
andProof of Lemma 10. By (7.5) in Lemma 7 and by Lemma 8, (7.13) and(7.14) in Lemma 9 hold with uj and uj in place of u and u~, respectively.
Thus by Lemma 9 we havewhere, by (7.4) in Lemma 7, Lemma 8 and (7.19) we have

246

It follows from (7.24) that

where by (7.25) the error term in the exponent is

Moreover, the innermost term in the main term in the exponent in (7.26)
can be rewritten as

Writing here

the last expression becomes a polynomial, in the variables a and x, of theformh, then clearlyi.e., d(h, t, k) is independent of 1~ for 1~ &#x3E; h. Thus we may write

247

In particular, computing these numbers d(h, i) for h 6 we obtain

Using this notation, by (7.28) and (7.29) for ~a~the exponent in (7.26) becomesq the main term in

where Lh and, in particular, L2, L3, L4, L5 and L6 are defined as in thetheorem, and by (1.7) and (7.30), the first term is

248

It follows from (7.2), (7.26), (7.27), (7.31) and (7.32) that for
integrand in J is

By (5.1) and Lemma 2, for mwhile for m &#x3E;

71 thein the error term we haveby k &#x3E; 3 we haveBy (1.15), both upper bounds in (7.34) and (7.35) are o(l) and thus in(7.33) we may write

with

249

Next in (7.33) we write

whereThen for

we have

By using (5.1) and Lemma 2, it follows in the same way as in (7.34) and(7.35) thatandBy (1.15), it follows from (7.41) and (7.42) that(uniformly forwithThus the second factor in (7.38) can be written as

250

Moreover, the main term in (7.44) isv-2+2.

Let max* denote the maximum
Consider here the terms with ttaken over the integers v, t, ~1, ..., ht with 1 v 1~2, 1 t k, t
1
3 h 1, ... , ht
k, h 1 + + ht = v . Then uniformly for lal1 q, by ( 5 .1 )isand Lemma 2 each of the terms with tv-2+2 ,...

v-2+2

so that for mwhile for m &#x3E;

Hnll2,,

251

at theThus in (7.46) we may restrict ourselves to terms withexpense of a small error term:

withNote that the summation over t is empty unlesswhenceso that the summation

can be replaced byfrom (7.33), (7.36), (7.37), (7.38), (7.39),
itfollows
Defining Z" by (7.22),(7.43), (7.44), (7.45), (7.49), (7.50) and (7.51) that uniformly for ~a~ q
the integrand in J1 iswhereand this completes the proof of Lemma 10.

*252

8. ESTIMATE OF JlLemma 11. We havewhereProof of Lemma 11. By Lemma 10 we havewith Eo satisfying (7.23). It remains to estimate the integrals

If v (&#x3E; 0) is odd then the integrand is odd and thus

If v

=2t is even, then substituting x=(2L2 )1/2 7ra we obtain

withIt is easy to see by induction (using integration by parts) that we have

253

Moreover, we havesay. By (1.16), (5.1) and Lemma 3 we have

If now we fix c* to be large enough in terms of k (and so that (5.9) shouldalso hold) then it follows thatBy (8.3), (8.4), (8.5), (8.6), (8.7) and (8.8) (and since Zv
1 v k2 by the proof of Lemma 10) we have=0(l) for

254

~whence, by (1.16), (7.23) and (8.8), (8.1) and (8.2) follow.

09. COMPLETION OF THE PROOF OF THEOREM 1

The result follows from (1.9), (5.2), Lemma 6 and Lemma 11. In particular, the formulas for L2, ..., L6 are obtained from (1.10) by using the numbersd(x, y) computed in the proof of Lemma 10, while (1.17) and (1.18) are thek
6 special cases of (1.12) and (1.14), respectively.
=REFERENCES[1] J. Dixmier, J.-L. Nicolas, Partitions without small parts. Number theory, Vol. I (Budapest,1987), 9-33, Colloq. Math. Soc. János Bolyai 51, North-Holland, Amsterdam, 1990.
[2] J. Dixmier, J.-L. Nicolas, Partitions sans petits sommants. A tribute to Paul Erdõs, 121-152,Cambridge Univ. Press, Cambridge, 1990.

[3] P. Erdõs, J.-L. Nicolas, M. Szalay, Partitions into parts which are unequal and large. Numbertheory (Ulm, 1987), 19-30, Lecture Notes in Math., 1380, Springer, New York-Berlin, 1989.
[4] P. Erdõs, M. Szalay, On the statistical theory of partitions. Topics in classical number theory,Vol. I, II (Budapest, 1981), 397-450, Colloq. Math. Soc. János Bolyai 34, North-Holland,Amsterdam-New York, 1984.

[5] G. Freiman, J. Pitman, Partitions into distinct large parts. J. Australian Math. Soc. Ser. A
57 (1994), 386-416.
[6] G. Szekeres, An asymptotic formula in the theory of partitions. Quart. J. Math. Oxford 2(1951), 85-108.
[7] G. Szekeres, Some asymptotic formulae in the theory of partitions II. Quart. J. Math. Oxford
4 (1953), 96-111.
J.-L. NICOLAS
Institut Girard Desargues, UMR 5028
Mathematiques, Bltiment 101
Universit6 Claude Bernard (Lyon 1)
F-69622 Villeurbanne Cedex, FRANCE

jlnicolaQin2p3.fr
A. SARK6ZY

Department of Algebra and Number Theory
E6tv6s Lorând University
Kecskem6ti utca 10-12
H-1088 Budapest, Hungary
E-mail : sarkozyQcs . elte . hu