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On a Generalization of Waring's Formula
ADVANCES IN APPLIED MATHEMATICS ARTICLE NO.
19, 450]452 Ž1997.
AM970545
On a Generalization of Waring’s Formula Jiang Zeng Departement de Mathematique, Uni¨ ersite´ Louis Pasteur, ´ ´ 67084 Strasbourg Cedex, ´ France Received January 1, 1996; accepted December 30, 1996
Konvalina w J. Combine Theory Ser. A 75 Ž1996., 281]294x gives a generalization of Waring’s formula. Here we show that Konvalina’s extension is a straightforward Q 1997 Academic Press consequence of the Waring formula itself.
Let x 1 , x 2 , . . . , x m be m independent variables. Recall w2x that for each k G 1 the kth elementary symmetric function and the kth power sum are given, respectively, by ek [ ek Ž x1 , x 2 , . . . , x m . s
Ý
i1-i 2- ??? -i k
x i1 x i 2 ??? x i k ,
m
pk [ pk Ž x 1 , x 2 , . . . , x m . s
Ý
x ik .
is1
A partition is a finite sequence l s Ž l1 , l2 , . . . , l r . of positive integers in decreasing order: l1 G l2 G ??? G l r ) 0. The l i are parts of l and the r is called the length of l, denoted by l Ž l.. The sum of the parts of l is denoted by < l < s l1 q l2 q ??? ql r . If < l < s n we say that l is a partition of n. The multiplicity of i in l, denoted by m i Ž l., is the number of parts of l equal to i. For each partition l we note el s e1m 1Ž l. e2m 2 Ž l. ??? ,
pl s p1m 1Ž l . p 2m 2 Ž l . ??? ,
and Zl s Ł i G 1 i m i Ž l. ? m i Ž l.!. If l, m are partitions, we define l j m to be the partition whose parts are those of l and m. 450 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
WARING’S FORMULA
451
The inverse formula w2, p. 24x of
Ž y1. nyl l
Ž .
en s
Ý
Zl
< l
pl ,
Ž 1.
expressed as pn s
n
Ž . Ý Ž y1. nyl l l Ž l.
< l
ž
l Ž l. e m1 Ž l . , m 2 Ž l . , . . . , m m Ž l . l
/
Ž 2.
is usually referred to Waring’s formula w3x. The extension of Ž2. proposed by Konvalina Žin its corrected form. w1x reads e k Ž x 1n , x 2n , . . . , x mn . s Ž y1 .
k Ž nq1 .
Ý
< l
Al el ,
Ž 3.
where the coefficients Al are given by
Ž y1. l l yl p k Ł is1 mi Ž p . ! Ž .
Al s
Ý
Ý
n
is1
l Ž li .
=Ł
ž
l Ž li . m1 Ž l i . , . . . , m m Ž l i .
/
Ž .
.
Formula Ž3. can be immediately derived from Ž2. as follows. First note that if f Ž x 1 , x 2 , . . . , x m . is a symmetric function whose expression in terms of the pi s is g Ž p1 , p 2 , . . . ., then f Ž x 1n , x 2n , . . . , x mn . s g Ž pn , p 2 n , . . . . .
Ž 4.
It follows from Ž1. that ek Ž
x 1n ,
x 2n , . . . ,
x mn
.s
Ž y1. kyl p
Ž . l Žp .
Ý
Zp
Ł pk n .
is1
i
Ž 5.
Now plugging Waring’s formula Ž2. in each pk i n of the preceding formula yields Ž3.. Remark. In Konvalina’s original formula w1x the multiplicities m i Žp . were incorrectly stated as the multiplicities of l i in l due to a counting error in his involved proof.
452 COROLLARY 1.
JIANG ZENG
If k s m, then
Ž y1. m nq1 Al s Ž
.
½
1, 0,
if l s Ž m n . , otherwise.
Proof. Since e mŽ x 1n, x 2n, . . . , x mn . s e mn , the corollary follows then from the fundamental theorem on symmetric functions. Remark. In the k s m case, contrary to the statement in the corollary of w1x, the summation constraints of the preceding theorem do not reduce the partitions of mn to the only partition l s Ž m n .. In fact the terms corresponding to the other partitions have just cancelled each other.
REFERENCES 1. J. Konvalina, A generalization of Waring’s formula, J. Combin. Theory Ser. A 75 Ž1996., 281]294. 2. I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ 2nd ed., Clarendon Press, Oxford, 1995. 3. M. P. Macmahon, ‘‘Combinatory Analysis,’’ Chelsea, New York, 1960.
19, 450]452 Ž1997.
AM970545
On a Generalization of Waring’s Formula Jiang Zeng Departement de Mathematique, Uni¨ ersite´ Louis Pasteur, ´ ´ 67084 Strasbourg Cedex, ´ France Received January 1, 1996; accepted December 30, 1996
Konvalina w J. Combine Theory Ser. A 75 Ž1996., 281]294x gives a generalization of Waring’s formula. Here we show that Konvalina’s extension is a straightforward Q 1997 Academic Press consequence of the Waring formula itself.
Let x 1 , x 2 , . . . , x m be m independent variables. Recall w2x that for each k G 1 the kth elementary symmetric function and the kth power sum are given, respectively, by ek [ ek Ž x1 , x 2 , . . . , x m . s
Ý
i1-i 2- ??? -i k
x i1 x i 2 ??? x i k ,
m
pk [ pk Ž x 1 , x 2 , . . . , x m . s
Ý
x ik .
is1
A partition is a finite sequence l s Ž l1 , l2 , . . . , l r . of positive integers in decreasing order: l1 G l2 G ??? G l r ) 0. The l i are parts of l and the r is called the length of l, denoted by l Ž l.. The sum of the parts of l is denoted by < l < s l1 q l2 q ??? ql r . If < l < s n we say that l is a partition of n. The multiplicity of i in l, denoted by m i Ž l., is the number of parts of l equal to i. For each partition l we note el s e1m 1Ž l. e2m 2 Ž l. ??? ,
pl s p1m 1Ž l . p 2m 2 Ž l . ??? ,
and Zl s Ł i G 1 i m i Ž l. ? m i Ž l.!. If l, m are partitions, we define l j m to be the partition whose parts are those of l and m. 450 0196-8858r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
WARING’S FORMULA
451
The inverse formula w2, p. 24x of
Ž y1. nyl l
Ž .
en s
Ý
Zl
< l
pl ,
Ž 1.
expressed as pn s
n
Ž . Ý Ž y1. nyl l l Ž l.
< l
ž
l Ž l. e m1 Ž l . , m 2 Ž l . , . . . , m m Ž l . l
/
Ž 2.
is usually referred to Waring’s formula w3x. The extension of Ž2. proposed by Konvalina Žin its corrected form. w1x reads e k Ž x 1n , x 2n , . . . , x mn . s Ž y1 .
k Ž nq1 .
Ý
< l
Al el ,
Ž 3.
where the coefficients Al are given by
Ž y1. l l yl p k Ł is1 mi Ž p . ! Ž .
Al s
Ý
Ý
n
is1
l Ž li .
=Ł
ž
l Ž li . m1 Ž l i . , . . . , m m Ž l i .
/
Ž .
.
Formula Ž3. can be immediately derived from Ž2. as follows. First note that if f Ž x 1 , x 2 , . . . , x m . is a symmetric function whose expression in terms of the pi s is g Ž p1 , p 2 , . . . ., then f Ž x 1n , x 2n , . . . , x mn . s g Ž pn , p 2 n , . . . . .
Ž 4.
It follows from Ž1. that ek Ž
x 1n ,
x 2n , . . . ,
x mn
.s
Ž y1. kyl p
Ž . l Žp .
Ý
Zp
Ł pk n .
is1
i
Ž 5.
Now plugging Waring’s formula Ž2. in each pk i n of the preceding formula yields Ž3.. Remark. In Konvalina’s original formula w1x the multiplicities m i Žp . were incorrectly stated as the multiplicities of l i in l due to a counting error in his involved proof.
452 COROLLARY 1.
JIANG ZENG
If k s m, then
Ž y1. m nq1 Al s Ž
.
½
1, 0,
if l s Ž m n . , otherwise.
Proof. Since e mŽ x 1n, x 2n, . . . , x mn . s e mn , the corollary follows then from the fundamental theorem on symmetric functions. Remark. In the k s m case, contrary to the statement in the corollary of w1x, the summation constraints of the preceding theorem do not reduce the partitions of mn to the only partition l s Ž m n .. In fact the terms corresponding to the other partitions have just cancelled each other.
REFERENCES 1. J. Konvalina, A generalization of Waring’s formula, J. Combin. Theory Ser. A 75 Ž1996., 281]294. 2. I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ 2nd ed., Clarendon Press, Oxford, 1995. 3. M. P. Macmahon, ‘‘Combinatory Analysis,’’ Chelsea, New York, 1960.