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On The Number of Partitions of N Without a Given Subsum (I)
Discrete Mathematics 75 (1989) 155-166 North-Holland
155
ON THE NUMBER OF PARTITIONS OF n WITHOUT A GIVEN SUBSUM (I) P. ERDOS,? J.L. NICOLASS and A. SARKOZYt* f A Magyar Tudomdnyos Akade'mia, Matemtikai Kutato inthete, Realtanoh u. 13-15, Pf. 127, H-1364 Budapest, Hungary. $ De'partement de Mathhatiques, bat. 101, Universite' Claude Bernard, Lyon 1, F-6%22 Villeurbanne Cedex, France
1. Introduction
Let us denote by p ( n ) the number of unrestricted partitions of n, by r(n, m) the number of partitions of n whose parts are at least m, and by R(n, a ) the number of partitions of n:
+ - + subsums n,, + . . +nil are n =n,
whose
*
*
nt,
all different from a. Furthermore, if d = {a,, . . . ,ak}, we denote by r(n, d)the number of partitions of n with no parts belonging to d. Let us consider now partitions of n for which each part is allowed to occur at most once. In that case the above notations will be changed for q ( n ) , p(n, m), Q(n, a ) , d n , 4. Clearly we have: *
r(n, m)= r(n, {1,2,
. . . , m - 1))
R(n, a ) a r(n, a + 1) R(n, a ) 3 r(n, {1,2, . . . , La/21, a } ) where Ix/21 denotes the integral part of x. In [4], the following estimation is given for R(n, a ) : when a is fixed, and n tends to infinity,
+
where V ( a ) = [a/2 11, and u(a) E N. The value of u ( a ) is computed for 1s a ss 20, and it does not seem easy to get a simple formula for u(a). The results are u(l)= 1, u(2)=4, u ( 3 ) = 3 , u(4)= 16, and u is increasing from a = 3 to * Research partially supported by Hungarian National Foundation for Scientific Research, grant no. 1811, and by C.N.R.S, Grew Calcul Formel and PRC Math.-Info. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)
P. Erd&
156
el al.
a = 20. Moreover, J. Dixmier gives the following inequalities: for a even for a odd
( [ a / 3 ]- l)!a0'6+3s u ( a ) s 2"'2a!/(a/2- l)!
([a/3] - l)!
s u ( a ) s 2"Qa!/(\a/2])!
(4)
(5)
It follows from (3), the definition of t,O, and the behaviour of u, that for n large enough, R(n, 1) > R(n, 2) > R(n, 3) > R ( n , 4)
(6)
and for a =2b, 2 s b S 9 , R(n, 26
+ 2) < R(n, 26) < R(n, 26 + 1)< R(n, 26 - 1).
At the end of the paper, a table of R ( n , a ) is given. It has been calculated by J. Dixmier, H. Epstein and O.E. Lanford, using the induction formula.
f ( n ,p , a)=
C f ( n - i, i, d u SB - i ) .
i =sp
Here, f (n, p , d)denotes the number of partitions of n in parts s p such that no subsum belongs to d,and d - i = { a - i ; a E d,a - i > O } . It has been independently calculated by F. Morain and J.P. Massias. They have used computer algebra systems MAPLE and MACSYMA to compute polynomials mentioned by Diximier (cf. [4], 4.3 and 4.10). Unfortunately these polynomials are of degree ( ( a + l)(a + 2)/2) - 2, and it is not easy to deal with them for large values of a. As observed in [4], R(n, 2) < R(n, 3) for 1 0 s n S 106, which contradicts (6). But (6) is true only for n large enough. The aim of this paper is to study R(n, a ) for a depending on n, and smaller where ;11, is a small positive constant. The tools for that are an than estimation for r(n, d )(cf. Lemma 2 below), and inequalities involving R(n, a), extending (1) and (2). We shall prove the following result.
at/;;,
Theorem 1. There exists &,> 0, such that uniformly for 1 == a s LOG, we have, when n goes to infinity,
where ya = $ if a is odd, and, i f a is even, ya = 4
a + log3 - 2 log2 + c -log = a
+0.79.
-
log a +cwhere c is a f i e d constant. a
Partitions of n without a given subsum
157
Let us observe that, when a goes to infinity, (4) or (5) gives
( - l + F+3 o(1))a
-
< 1og u(a) - ;a
log a s (log 2 - ;+ o(1))a
(7)
while (3), (i) and (ii) yield that
- y,a
+ o(a) s log u(a) -
$2
log a s o(a)
which is better than (7) except for the lower bound when a is even. 6 6 a in an other paper, by a different method, We intend to treat the case &, which will give also an estimation for Q(n, a). For this quantity, we here give only a lower bound.
Theorem 2. There exists Al > 0, such that, uniformly for 1s a s A,fi,
we have:
We thank very much J. Dixmier for several interesting remarks.
2. Preliminary Results Let us first recall the definition of the mth Bessell polynomial y,(x); (cf. [9]): Y&) = 1 yrn(x) = (1+ m)Ym-l(x) -I-x2~:,-l(x)-
(9)
From that definition, it is easy to see that, if we set ~ ( x =) (exp (fi))/fi,
then we have (cf. [5], Lemma 1)
Furthermore, it follows from (9) that ym(0) = 1.
Lemma 1. For m odd, the function x+ym(x) is increasing for x its zero a,,,satisfies
~ ] - m , +m[,
and
Y
m+l
where y is a constant satisfying 1.5 < y < 1.51. For m even, ym(x) is decreasing on ]-m, a;[ and increasing on I&,
-
m + 0.77
Y
m + 1.61.
Proof. This is proved in [l] and 121. 0
+w[, and
P. EraE et al.
158
Lemma 2. Let us define P, by
Then we have
Proof. It is known that ym(x) satisfies the differential equation (cf. [9], p. 7) x2y"
+ 2(x + 1)y' - m(m + 1)y = 0.
If we set w ( x ) = y,(-x), xZw"
(13)
it satisfies
+ 2(x - 1)w' - m(m + l ) w = 0.
Now we set Y = yw. It is known that Y satisfies a linear differential equation, and
with some calculation this equation writes:
x 4 Y + 6x3Y" - ((4m(m + 1 ) - 6)x2 + 4 ) Y ' - 4 m ( m + 1)xY = 0.
(15)
We can easily check that Y satisfies (15), by calculating Y" and Y"' in terms of
yw, y ' w , y w ' , y ' w ' by (13) and (14).
Now, we are looking for polynomial solutions of (15). It turns out that these solutions are of the form cP,(x'), and considering x = 0 yields Lemma 2. El We are very pleased to thank A. Salinier €or this proof of Lemma 2. This result the is somewhat curious. We would expect that, in the product y,(x)y,,,(-x) coefficient of x2 is a polynomial of degree 4k in m. Indeed, it follows from (9), cf. 191, p. 13, that y,(x) = 1 +
with
C aim)xk m
k=l
k
Lemma 3. For x such that 0 s m s l / a , we have
Proof. From the obvious inequality (m - i + l ) ( m
+ i) S ( m + l)m, (16) implies
Partitions of n without a given subsurn
which gives yrn(x>s exp(
m(m2+
"x).
159
(17)
Now, let us write the polynomial P,, defined in Lemma 2 , in the form rn
Then we have
Thus, the absolute value of the general term of yrn(x)yrn(-x), that is (dkxzk(is decreasing, and, as it alternates in sign, for m s l / a , we have 1-
m(m + 1) x2 Gyrn(x)yrn(-x) = Prn(x2) 1, 2
which, with (17) completes the proof of Lemma 3. 0 More accurate estimations have been obtained by M. Chellali, using Agarwal's integral representation yrn(X)
+
=n . [ t n ( 1
dt
and the saddle point method (cf. [3]).
Proposition. There exists k2> 0 such that, if d = {al, . . ,ak} satisfies s = a l + a, + ak 6 A2n, then, when n ten& to infinity, we have
- +
Proof. It is very similar to the proof of Theorem 1 of [5].First we introduce the operator D'"). Let f : Iw +R, m 2 1, u l , . . . , urnbe positive. We set D(l)(u1;f,x ) =f(x) -f(x - u1)
D(rn)(U1,. . . , u,;f, x ) = D ( m - l ) ( U ] , . . . , u,-,;f, x )
-p
- 1 )
( ~ 1., . * urn-,;f, x - urn). 9
From the generating functions, we observe
r(n, {al, . . . ,ak}) = ~ ( ~ ) ( a. .l ., ,ak;p, n).
P. Erd& et al.
160
Now, with F ( x ) = (exp(fi))/fi, the classical result of Hardy and Ramanujan can be written (cf. [5]) as follows:
c3 ht/Z
p ( n ) = -F'(CZ(n - 1/24)) + f i ( n )
(19)
with C = n m , and
Furthermore, using Lemma 3 of [5], (18) and (19) give
with
n -s
=S(5 s
n.
(21)
- 1/24)). The function x--, exx( kp+ 2(1V/23 is We now use (10) to estimate F(k+')(C2(E
-
increasing for x z= (k + 2),. As s = a , + . * + a k a k(k for A, small enough, from (21) we have plies k s
6,
+ 1 ) / 2 3 k2/2, which
Now let us turn to the proof of (i). By (22), the main term of (20), is at most
For A2 small enough, we have
~
-1
%
Y
-
and thus Lemma 1 gives
-I Y k + l ( Z i 7 = g T E )
1.
Using the estimation
the main term of (20) is at most
z
im-
Partitions of n without a given subsum
161
To complete the proof of (i), it remains to check that the error term of (20) is included in the above error term. First, (24) is
>> k ! by Stirling’s formula. So it is enough to show that ( 4efi xrsexp(gfi). But the left hand side of the above inequality is an increasing function on k , for 4 k s - fi,and we know that k C fiC To conclude, we observe that for C A2 small enough, we have:
a.
In order to prove (ii), first we apply Lemma 3 , to obtain
which with (23) yields
Then we observe that and
e x p ( C V K G - 7 ) = exp(Cfi ( n - s - l ) ( k + z ) n= exp(
+~(s/fi)) (log n + O ( s / n ) ) )
- n(“’)’’ exp(O(s/fi)) since k = O(fi). Furthermore, by ( l o ) , (22) and (25), the main term of (20) is at least
The end of the proof of (ii) goes in the same way as for (i). 0
Remark. A similar proposition is given in [7] in the case of restricted partitions.
A more general estimation is given by J. Herzog (cf. [lo] and [ l l ] ) ,using a Tauberian theorem.
P. E&"s et al.
162
3. The upper bound in Theorem 1
---
First let us say that a partition n = n1+ + n, of n represents a if there is a < ij 6 t which is equal to a. Thus R(n, a) counts subsum nil,. . . , ni,, 1S il < the number of partitions of n which do not represent a. Clearly if b < a , b and a - b cannot be together parts of a partition which does not represent a. Let us suppose first that a is odd. From the above remark, we deduce that for all integers, i, with 16 i 6 [ a / 2 ] , at most one of i and a - i can be a part, and thus
- -
where in the summation E~ E (0, l}. Now we apply our proposition, with k = W ( a ) , and obtain that
S S E ~ and ~ we ~ ~ ~ ~
But this summation is exactly
a
n (i +
Lon1
( a - i)) = a'(a)
i=l
which proves (i) for a odd. When a is even, the part a / 2 can occur but only once. Thus we have 1G' {iE,@
- i)l-E,
i=l
where the first summation counts partitions without any part equal to a/2, and the second counts partitions with one part equal to a / 2 . For the second sum we obtain the upper bound
But, as already observed, the function x-,expfi/xk is increasing for x and for &, small enough, the expression between brackets is smaller than
3 k2,
Partitions of n without a given subsum
163
So, the second sum in (27) is not bigger than the first one, which was already estimated when a is odd. This completes the proof of (i). 0
4. The lower bound in Theorem 1 Let us suppose first that a is odd. Then R(n, a)>rr(n, {1,3,5,. . . , a } )
and, observing that @ ( a )= (a + 1)/2, by the Proposition we have
By Stirling's formula, (2u)!/2"u! 2 ~ " 2 ~ e -and " , since @ ( a )3 a/2, we obtain (ii). Let us suppose now that a is even. In fact, the following reasoning works also for a odd, but it gives a worse estimation than the preceding one. For real numbers x and y, let us denote the set of integers belonging to the real interval ]x, y [ by ]x - . y [ . We set d = [1* ~ / 3U ] [ ~ / *2* * h / 3 ] U { a } . +
--
Then, it is not difficult to see that
R(n, a ) 3 r(n, d) (which is slightly bettter than (2)), and considering the three possible cases a = 0,2,4 mod 6, that card d = @(a).By the proposition, we get
Using Stirling's formula in the form [u O(l)]! = uUe-"exp(O(1og u ) )
+
we obtain (ii) with an effectively computable constant c. 0
5. Proof of Theorem 2 We consider now only partitions without any repetition, and we look at a subset of [l - - (a - l)], say d with the following property:
-
[ - - - i]belongs to d for each j €1; - - ;[, there are 3 possibilities: no element j E 1
j E d and a -j @ d,j 6 d and a - j E d,j 6 d and a - j 6 d
for each j E
[f - - a - 11, there are 2 possibilities, j *
E d or j 6 d.
P. Era% et al.
164
For any such d,we have: Card d s c,a2. How many such d ' s are there? As
[
C a r d ( ] ~ . . ~ ~ [ ) 2 ~ and - 1 Card( -. 2a 3
- - (a - l)]) 3 , 12
there are more than
2
3(i1/6)-1 013
such sets d.Further, to build a partition of n, we choose such a set d ,and we complete by a partition of n-Card d,without any part smaller than a 1. Thus, since p(n, rn) is non-decreasing in n (cf. [S]),
+
Q(n, a ) 3 3a'6-12013p(n - c ~ u ' ,u + 1).
Using Theorem 1 of [8], which gives p(n, r n ) 2 q ( n ) / 2 " - ' , estimation
and the classical
Table of R(n,-a) 2
3
4
5
6 1
101
135 176
385 490 627 792 1002 1255 1575 1958
I
I
I I I
I
I I
I
5 7 9 11 15 19 20 23 34 30 30 41 32 38 46 51 -70 Be 58 46
-72 10: 70 88 137 105 109 165 119 133 21( 156 161 25: 177 198 32( 228 240 38: 262 ~
~
..
7
8
9
1
0
1
1
1
1
2
Partitions of n without a given subsum
165
Table of R(n, a) (continued).
which implies
we obtain Theorem 2. Cl
References M.Chellali and A. Salinier, Sur les zeros des polynbmes de Bessel, CRAS, t. 305, Sene 1 (1987) 765-767.
M. Chellali, Sur les zeros des polynbmes de Bessel II, CRAS, t. 307, Stne 1 (1988) 547-550. M.Chellali, Sur les ZRros des polynbmes de Bessel 111, CRAS, t. 307,S&ie 1 (1988) 651-654. J. Dixmier, Sur les sous sommes d‘une partition, prepnnt de I’IHES (1987) and to appear in Memoires SOC.Math. France. J. Dixmier and J.L.Nicolas, Partitions without small parts, to be published in the Proceedings of Colloquium in Number Theory, Budapest (July 1987).
P. Era%
166
et al.
[6j P. Erdiis and M. Szalay, On some problems of J. DBnes and P. Turan, Studies in pure Mathematics to the memory of P. Turan, Editor P. Erdiis, Budapest (1983) 187-212 [7] P. Erd& and M. Szalay, On some problems of the statistical theory of partitions, to be published in the Proceedings of Colloquium in Number Theory, Budapest (July 1987). 181 P. Erdtjs, J.L. Nicolas and M. Szalay, Partitions into parts which are unequal and large, to be
published in the Proceedings of Journ6es ArithmCtiques, Ulm, Germany (Springer Verlag Lecture Notes). [9] E. Grosswald, Bessel polynomials, Lecture Notes in Mathematics No. 698 (Springer Verlag, 1978).
[lo] J. Herzog, GleichmWige asymptotische Formeln fiir parameterabhangige Partition funktionen, Thesis of Univ. J.F. Goethe, Frankfurt am Main (1987). [ll] J. Herzog, O n partitions into distinct parts a y , preprint.
155
ON THE NUMBER OF PARTITIONS OF n WITHOUT A GIVEN SUBSUM (I) P. ERDOS,? J.L. NICOLASS and A. SARKOZYt* f A Magyar Tudomdnyos Akade'mia, Matemtikai Kutato inthete, Realtanoh u. 13-15, Pf. 127, H-1364 Budapest, Hungary. $ De'partement de Mathhatiques, bat. 101, Universite' Claude Bernard, Lyon 1, F-6%22 Villeurbanne Cedex, France
1. Introduction
Let us denote by p ( n ) the number of unrestricted partitions of n, by r(n, m) the number of partitions of n whose parts are at least m, and by R(n, a ) the number of partitions of n:
+ - + subsums n,, + . . +nil are n =n,
whose
*
*
nt,
all different from a. Furthermore, if d = {a,, . . . ,ak}, we denote by r(n, d)the number of partitions of n with no parts belonging to d. Let us consider now partitions of n for which each part is allowed to occur at most once. In that case the above notations will be changed for q ( n ) , p(n, m), Q(n, a ) , d n , 4. Clearly we have: *
r(n, m)= r(n, {1,2,
. . . , m - 1))
R(n, a ) a r(n, a + 1) R(n, a ) 3 r(n, {1,2, . . . , La/21, a } ) where Ix/21 denotes the integral part of x. In [4], the following estimation is given for R(n, a ) : when a is fixed, and n tends to infinity,
+
where V ( a ) = [a/2 11, and u(a) E N. The value of u ( a ) is computed for 1s a ss 20, and it does not seem easy to get a simple formula for u(a). The results are u(l)= 1, u(2)=4, u ( 3 ) = 3 , u(4)= 16, and u is increasing from a = 3 to * Research partially supported by Hungarian National Foundation for Scientific Research, grant no. 1811, and by C.N.R.S, Grew Calcul Formel and PRC Math.-Info. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)
P. Erd&
156
el al.
a = 20. Moreover, J. Dixmier gives the following inequalities: for a even for a odd
( [ a / 3 ]- l)!a0'6+3s u ( a ) s 2"'2a!/(a/2- l)!
([a/3] - l)!
s u ( a ) s 2"Qa!/(\a/2])!
(4)
(5)
It follows from (3), the definition of t,O, and the behaviour of u, that for n large enough, R(n, 1) > R(n, 2) > R(n, 3) > R ( n , 4)
(6)
and for a =2b, 2 s b S 9 , R(n, 26
+ 2) < R(n, 26) < R(n, 26 + 1)< R(n, 26 - 1).
At the end of the paper, a table of R ( n , a ) is given. It has been calculated by J. Dixmier, H. Epstein and O.E. Lanford, using the induction formula.
f ( n ,p , a)=
C f ( n - i, i, d u SB - i ) .
i =sp
Here, f (n, p , d)denotes the number of partitions of n in parts s p such that no subsum belongs to d,and d - i = { a - i ; a E d,a - i > O } . It has been independently calculated by F. Morain and J.P. Massias. They have used computer algebra systems MAPLE and MACSYMA to compute polynomials mentioned by Diximier (cf. [4], 4.3 and 4.10). Unfortunately these polynomials are of degree ( ( a + l)(a + 2)/2) - 2, and it is not easy to deal with them for large values of a. As observed in [4], R(n, 2) < R(n, 3) for 1 0 s n S 106, which contradicts (6). But (6) is true only for n large enough. The aim of this paper is to study R(n, a ) for a depending on n, and smaller where ;11, is a small positive constant. The tools for that are an than estimation for r(n, d )(cf. Lemma 2 below), and inequalities involving R(n, a), extending (1) and (2). We shall prove the following result.
at/;;,
Theorem 1. There exists &,> 0, such that uniformly for 1 == a s LOG, we have, when n goes to infinity,
where ya = $ if a is odd, and, i f a is even, ya = 4
a + log3 - 2 log2 + c -log = a
+0.79.
-
log a +cwhere c is a f i e d constant. a
Partitions of n without a given subsum
157
Let us observe that, when a goes to infinity, (4) or (5) gives
( - l + F+3 o(1))a
-
< 1og u(a) - ;a
log a s (log 2 - ;+ o(1))a
(7)
while (3), (i) and (ii) yield that
- y,a
+ o(a) s log u(a) -
$2
log a s o(a)
which is better than (7) except for the lower bound when a is even. 6 6 a in an other paper, by a different method, We intend to treat the case &, which will give also an estimation for Q(n, a). For this quantity, we here give only a lower bound.
Theorem 2. There exists Al > 0, such that, uniformly for 1s a s A,fi,
we have:
We thank very much J. Dixmier for several interesting remarks.
2. Preliminary Results Let us first recall the definition of the mth Bessell polynomial y,(x); (cf. [9]): Y&) = 1 yrn(x) = (1+ m)Ym-l(x) -I-x2~:,-l(x)-
(9)
From that definition, it is easy to see that, if we set ~ ( x =) (exp (fi))/fi,
then we have (cf. [5], Lemma 1)
Furthermore, it follows from (9) that ym(0) = 1.
Lemma 1. For m odd, the function x+ym(x) is increasing for x its zero a,,,satisfies
~ ] - m , +m[,
and
Y
m+l
where y is a constant satisfying 1.5 < y < 1.51. For m even, ym(x) is decreasing on ]-m, a;[ and increasing on I&,
-
m + 0.77
Y
m + 1.61.
Proof. This is proved in [l] and 121. 0
+w[, and
P. EraE et al.
158
Lemma 2. Let us define P, by
Then we have
Proof. It is known that ym(x) satisfies the differential equation (cf. [9], p. 7) x2y"
+ 2(x + 1)y' - m(m + 1)y = 0.
If we set w ( x ) = y,(-x), xZw"
(13)
it satisfies
+ 2(x - 1)w' - m(m + l ) w = 0.
Now we set Y = yw. It is known that Y satisfies a linear differential equation, and
with some calculation this equation writes:
x 4 Y + 6x3Y" - ((4m(m + 1 ) - 6)x2 + 4 ) Y ' - 4 m ( m + 1)xY = 0.
(15)
We can easily check that Y satisfies (15), by calculating Y" and Y"' in terms of
yw, y ' w , y w ' , y ' w ' by (13) and (14).
Now, we are looking for polynomial solutions of (15). It turns out that these solutions are of the form cP,(x'), and considering x = 0 yields Lemma 2. El We are very pleased to thank A. Salinier €or this proof of Lemma 2. This result the is somewhat curious. We would expect that, in the product y,(x)y,,,(-x) coefficient of x2 is a polynomial of degree 4k in m. Indeed, it follows from (9), cf. 191, p. 13, that y,(x) = 1 +
with
C aim)xk m
k=l
k
Lemma 3. For x such that 0 s m s l / a , we have
Proof. From the obvious inequality (m - i + l ) ( m
+ i) S ( m + l)m, (16) implies
Partitions of n without a given subsurn
which gives yrn(x>s exp(
m(m2+
"x).
159
(17)
Now, let us write the polynomial P,, defined in Lemma 2 , in the form rn
Then we have
Thus, the absolute value of the general term of yrn(x)yrn(-x), that is (dkxzk(is decreasing, and, as it alternates in sign, for m s l / a , we have 1-
m(m + 1) x2 Gyrn(x)yrn(-x) = Prn(x2) 1, 2
which, with (17) completes the proof of Lemma 3. 0 More accurate estimations have been obtained by M. Chellali, using Agarwal's integral representation yrn(X)
+
=n . [ t n ( 1
dt
and the saddle point method (cf. [3]).
Proposition. There exists k2> 0 such that, if d = {al, . . ,ak} satisfies s = a l + a, + ak 6 A2n, then, when n ten& to infinity, we have
- +
Proof. It is very similar to the proof of Theorem 1 of [5].First we introduce the operator D'"). Let f : Iw +R, m 2 1, u l , . . . , urnbe positive. We set D(l)(u1;f,x ) =f(x) -f(x - u1)
D(rn)(U1,. . . , u,;f, x ) = D ( m - l ) ( U ] , . . . , u,-,;f, x )
-p
- 1 )
( ~ 1., . * urn-,;f, x - urn). 9
From the generating functions, we observe
r(n, {al, . . . ,ak}) = ~ ( ~ ) ( a. .l ., ,ak;p, n).
P. Erd& et al.
160
Now, with F ( x ) = (exp(fi))/fi, the classical result of Hardy and Ramanujan can be written (cf. [5]) as follows:
c3 ht/Z
p ( n ) = -F'(CZ(n - 1/24)) + f i ( n )
(19)
with C = n m , and
Furthermore, using Lemma 3 of [5], (18) and (19) give
with
n -s
=S(5 s
n.
(21)
- 1/24)). The function x--, exx( kp+ 2(1V/23 is We now use (10) to estimate F(k+')(C2(E
-
increasing for x z= (k + 2),. As s = a , + . * + a k a k(k for A, small enough, from (21) we have plies k s
6,
+ 1 ) / 2 3 k2/2, which
Now let us turn to the proof of (i). By (22), the main term of (20), is at most
For A2 small enough, we have
~
-1
%
Y
-
and thus Lemma 1 gives
-I Y k + l ( Z i 7 = g T E )
1.
Using the estimation
the main term of (20) is at most
z
im-
Partitions of n without a given subsum
161
To complete the proof of (i), it remains to check that the error term of (20) is included in the above error term. First, (24) is
>> k ! by Stirling’s formula. So it is enough to show that ( 4efi xrsexp(gfi). But the left hand side of the above inequality is an increasing function on k , for 4 k s - fi,and we know that k C fiC To conclude, we observe that for C A2 small enough, we have:
a.
In order to prove (ii), first we apply Lemma 3 , to obtain
which with (23) yields
Then we observe that and
e x p ( C V K G - 7 ) = exp(Cfi ( n - s - l ) ( k + z ) n= exp(
+~(s/fi)) (log n + O ( s / n ) ) )
- n(“’)’’ exp(O(s/fi)) since k = O(fi). Furthermore, by ( l o ) , (22) and (25), the main term of (20) is at least
The end of the proof of (ii) goes in the same way as for (i). 0
Remark. A similar proposition is given in [7] in the case of restricted partitions.
A more general estimation is given by J. Herzog (cf. [lo] and [ l l ] ) ,using a Tauberian theorem.
P. E&"s et al.
162
3. The upper bound in Theorem 1
---
First let us say that a partition n = n1+ + n, of n represents a if there is a < ij 6 t which is equal to a. Thus R(n, a) counts subsum nil,. . . , ni,, 1S il < the number of partitions of n which do not represent a. Clearly if b < a , b and a - b cannot be together parts of a partition which does not represent a. Let us suppose first that a is odd. From the above remark, we deduce that for all integers, i, with 16 i 6 [ a / 2 ] , at most one of i and a - i can be a part, and thus
- -
where in the summation E~ E (0, l}. Now we apply our proposition, with k = W ( a ) , and obtain that
S S E ~ and ~ we ~ ~ ~ ~
But this summation is exactly
a
n (i +
Lon1
( a - i)) = a'(a)
i=l
which proves (i) for a odd. When a is even, the part a / 2 can occur but only once. Thus we have 1G' {iE,@
- i)l-E,
i=l
where the first summation counts partitions without any part equal to a/2, and the second counts partitions with one part equal to a / 2 . For the second sum we obtain the upper bound
But, as already observed, the function x-,expfi/xk is increasing for x and for &, small enough, the expression between brackets is smaller than
3 k2,
Partitions of n without a given subsum
163
So, the second sum in (27) is not bigger than the first one, which was already estimated when a is odd. This completes the proof of (i). 0
4. The lower bound in Theorem 1 Let us suppose first that a is odd. Then R(n, a)>rr(n, {1,3,5,. . . , a } )
and, observing that @ ( a )= (a + 1)/2, by the Proposition we have
By Stirling's formula, (2u)!/2"u! 2 ~ " 2 ~ e -and " , since @ ( a )3 a/2, we obtain (ii). Let us suppose now that a is even. In fact, the following reasoning works also for a odd, but it gives a worse estimation than the preceding one. For real numbers x and y, let us denote the set of integers belonging to the real interval ]x, y [ by ]x - . y [ . We set d = [1* ~ / 3U ] [ ~ / *2* * h / 3 ] U { a } . +
--
Then, it is not difficult to see that
R(n, a ) 3 r(n, d) (which is slightly bettter than (2)), and considering the three possible cases a = 0,2,4 mod 6, that card d = @(a).By the proposition, we get
Using Stirling's formula in the form [u O(l)]! = uUe-"exp(O(1og u ) )
+
we obtain (ii) with an effectively computable constant c. 0
5. Proof of Theorem 2 We consider now only partitions without any repetition, and we look at a subset of [l - - (a - l)], say d with the following property:
-
[ - - - i]belongs to d for each j €1; - - ;[, there are 3 possibilities: no element j E 1
j E d and a -j @ d,j 6 d and a - j E d,j 6 d and a - j 6 d
for each j E
[f - - a - 11, there are 2 possibilities, j *
E d or j 6 d.
P. Era% et al.
164
For any such d,we have: Card d s c,a2. How many such d ' s are there? As
[
C a r d ( ] ~ . . ~ ~ [ ) 2 ~ and - 1 Card( -. 2a 3
- - (a - l)]) 3 , 12
there are more than
2
3(i1/6)-1 013
such sets d.Further, to build a partition of n, we choose such a set d ,and we complete by a partition of n-Card d,without any part smaller than a 1. Thus, since p(n, rn) is non-decreasing in n (cf. [S]),
+
Q(n, a ) 3 3a'6-12013p(n - c ~ u ' ,u + 1).
Using Theorem 1 of [8], which gives p(n, r n ) 2 q ( n ) / 2 " - ' , estimation
and the classical
Table of R(n,-a) 2
3
4
5
6 1
101
135 176
385 490 627 792 1002 1255 1575 1958
I
I
I I I
I
I I
I
5 7 9 11 15 19 20 23 34 30 30 41 32 38 46 51 -70 Be 58 46
-72 10: 70 88 137 105 109 165 119 133 21( 156 161 25: 177 198 32( 228 240 38: 262 ~
~
..
7
8
9
1
0
1
1
1
1
2
Partitions of n without a given subsum
165
Table of R(n, a) (continued).
which implies
we obtain Theorem 2. Cl
References M.Chellali and A. Salinier, Sur les zeros des polynbmes de Bessel, CRAS, t. 305, Sene 1 (1987) 765-767.
M. Chellali, Sur les zeros des polynbmes de Bessel II, CRAS, t. 307, Stne 1 (1988) 547-550. M.Chellali, Sur les ZRros des polynbmes de Bessel 111, CRAS, t. 307,S&ie 1 (1988) 651-654. J. Dixmier, Sur les sous sommes d‘une partition, prepnnt de I’IHES (1987) and to appear in Memoires SOC.Math. France. J. Dixmier and J.L.Nicolas, Partitions without small parts, to be published in the Proceedings of Colloquium in Number Theory, Budapest (July 1987).
P. Era%
166
et al.
[6j P. Erdiis and M. Szalay, On some problems of J. DBnes and P. Turan, Studies in pure Mathematics to the memory of P. Turan, Editor P. Erdiis, Budapest (1983) 187-212 [7] P. Erd& and M. Szalay, On some problems of the statistical theory of partitions, to be published in the Proceedings of Colloquium in Number Theory, Budapest (July 1987). 181 P. Erdtjs, J.L. Nicolas and M. Szalay, Partitions into parts which are unequal and large, to be
published in the Proceedings of Journ6es ArithmCtiques, Ulm, Germany (Springer Verlag Lecture Notes). [9] E. Grosswald, Bessel polynomials, Lecture Notes in Mathematics No. 698 (Springer Verlag, 1978).
[lo] J. Herzog, GleichmWige asymptotische Formeln fiir parameterabhangige Partition funktionen, Thesis of Univ. J.F. Goethe, Frankfurt am Main (1987). [ll] J. Herzog, O n partitions into distinct parts a y , preprint.