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Some experiments with complete and elementary symmetric functions

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Abstract

The complete and elementary symmetric functions are special cases of Schur functions. It is well-known that the Schur functions can be expressed in terms of complete or elementary symmetric functions using two determinant formulas: Jacobi–Trudi identity and Nägelsbach–Kostka identity. In this paper, we study new connections between complete and elementary symmetric functions.
Some experiments with complete and elementary
symmetric functions
Mircea Merca
Department of Mathematics, University of Craiova
Craiova, 200585 Romania
Abstract
The complete and elementary symmetric functions are special cases
of Schur functions. It is well-known that the Schur functions can be ex-
pressed in terms of complete or elementary symmetric functions using
two determinant formulas: Jacobi-Trudi identity and N¨agelsbach-Kostka
identity. In this paper, we study new connections between complete and
elementary symmetric functions.
Keywords: symmetric functions, Stirling numbers, determinants
MSC 2010: 05E05, 15A15
1 Introduction
For any positive integers nand k, the unsigned Stirling number of the first kind,
denoted by n
k,
is defined as the number of permutations of an n-set with kdisjoint cycles, and
the Stirling number of the second kind, denoted by
n
k,
is defined as the number of partitions of an n-set with knon-empty subsets.
We start with three experiments involving Stirling numbers of both kinds.
Experiment 1. Let
A(k)
n=i+k
j16i,j6n
and B(k)
n=i+k
j16i,j6n
mircea.merca@profinfo.edu.ro
1
be two square matrices where kis a non-negative integer. Using Maple to
compute the values of det A(k)
nand det B(k)
nfor various values of nand k, we
note the following two identities
det i+k
j16i,j6n
=k!n(1.1)
and
det i+k
j16i,j6n
=n!k.(1.2)
Experiment 2. Let
A(k)
n= i+k
j+k116i,j6n
and B(k)
n= i+k
j+k116i,j6n
be two square matrices where kis a positive integer. Using Maple to compute
the values of det A(k)
nand det B(k)
nfor various values of nand k, we note the
following two identities
det  i+k
j+k116i,j6n
=n+k
k(1.3)
and
det  i+k
j+k116i,j6n
=n+k
k.(1.4)
Experiment 3. We consider the square matrices
A(k)
n=i+k
j+ 116i,j6n
and B(k)
n=i+k
j+ 116i,j6n
where kis a positive integer. Using Maple to compute det A(k)
nfor various
values of nand k, we note that the sequences ndet A(2)
non>0,ndet A(3)
non>0,
...,ndet A(7)
non>0, respectively ndet A(8)
non>0are related to the sequences
A000225, A001240, A001241, A001242, A111886, A111887, respectively A111888
from OEIS [6]. In this way, we assume that the generating function for the se-
quence ndet A(k)
non>0is given by
1 + X
n>0
det A(k)
nxn=
k
Y
j=1 1k!
jx1
and
det i+k
j+ 116i,j6n
=k!n
k
X
j=1
(1)j11
jnk
j.(1.5)
2
On the other hand, we note that the sequences ndet B(k)
non>0,k∈ {2,3,4,5,6}
are related to the sequences A000254, A000424, A001236, A001237 and A001238
from OEIS [6]. Thus we assume that
det i+k
j+ 116i,j6n
= (n+ 1)!k1
n+1
X
j=1
(1)j11
jk1n+ 1
j.(1.6)
In this paper, we show that these experimental results are very special cases
of more general identities. In fact, it is well-known [4, 5] that the unsigned
Stirling numbers of the first kind are the elementary symmetric functions of the
numbers 1,2, . . . , n, i.e.,
n+ 1
n+ 1 k=ek(1,2,...,n) (1.7)
and the Stirling numbers of the second kind are the complete homogeneous
symmetric functions of the numbers 1,2,...,n, i.e.,
n+k
n=hk(1,2,...,n).(1.8)
Therefore, new connections between complete and elementary symmetric func-
tions will be studied in the paper.
2 Main results
We recall [3, Ch. I.3] that the Schur function sλ,n =sλ(x1, x2,...,xn) can be
defined as the ratio of two determinants as follows:
sλ(x1, x2,...,xn) =
det xλj+nj
i16i,j6n
det xnj
i16i,j6n
,
where λ= [λ1, . . . , λn] is an integer partition of length 6nand x1,...,xnare
independent indeterminates.
If λ= [1k], then sλ,n is the kth elementary symmetric function ek,n,
ek,n =ek(x1, x2,...,xn) = X
16i1<i2<...<ik6n
xi1xi2. . . xik.
When λ= [k], sλ,n is the kth complete homogeneous symmetric function hk,n ,
hk,n =hk(x1, x2,...,xn) = X
16i16i26...6ik6n
xi1xi2. . . xik.
In particular, we have e0,n = 1 and h0,n = 1. For k < 0 it is convenient to
define ek,n = 0 and hk,n = 0. When k > n we have ek,n = 0 and
hk(x1,...,xn) = hk(x1,...,xn,0,...,0
|{z }
kn
).
3
The generating function for the elementary symmetric function ek,n is
X
k>0
ek,ntk=
n
Y
i=1
(1 + xit)
and the generating function for the hk ,n is
X
k>0
hk,ntk=
n
Y
i=1
(1 xit)1.(2.1)
The following relations
ek,n = det (hij+1,n)16i,j6k
and
hk,n = det (eij+1,n)16i,j6k
are well-known. The first is a special case of Jacobi-Trudi identity [2, eq. 0.2]
and the second is a special case of N¨agelsbach-Kostka identity [2, eq. 0.3].
In this paper, we shall prove:
Theorem 2.1. Let aand kbe two nonnegative integers such that a6k. Then
det(eij+a,i1+k)16i,j6n= det(eij+a,k)16i,j6n.
Theorem 2.2. Let aand kbe two non-negative integers. Then
det(hij+a,j+k)16i,j6n= det(hij+a,n+k)16i,j6n.
Due to Giambelli’s determinant formulas [1], on the right hand side of
equations from Theorems 2.1 and 2.2, we have the Schur functions s[na],k and
s[an],n+krespectively.
The case a=kof Theorem 2.1 can be written as
Corollary 2.1. For k, n > 0,
det (eij+k(x1, x2,...,xi1+k))16i,j6n= (x1x2···xk)n.
By Theorem 2.2, with kreplaced by 0, we get
Corollary 2.2. For a, n > 0,
det (hij+a(x1, x2,...,xj))16i,j6n= (x1x2···xn)a.
4
Example. The case k= 2 of Corollary 2.1 is given by
e2,2e1,21
e3,3e2,3e1,31
e4,4e3,4e2,4e1,41
e5,5e4,5e3,5e2,5e1,5
e6,6e5,6e4,6e3,6e1,6
...
n×n
=en
2,2
and the case a= 2 of Corollary 2.2 can be written as
h2,1h1,21
h3,1h2,2h1,31
h4,1h3,3h2,3h1,41
h5,1h4,3h3,3h2,4h1,5
h6,1h5,3h4,3h3,4h2,5
...
n×n
=e2
n,n.
Replacing xiby iin these corollaries, we obtain the identities (1.1) and (1.2).
A generalization of the symmetry between the relations (1.3) and (1.4) can be
obtain replacing aby 1 in our theorems.
Corollary 2.3. For k, n > 0,
det (eij+1(x1, x2,...,xi1+k))16i,j6n=hn(x1, x2,...,xk),and
det (hij+1(x1, x2,...,xj1+k))16i,j6n=en(x1, x2,...,xn+k1).
Example. By Corollary 2.3 for k= 2, we have
e1,21
e2,3e1,31
e3,4e2,4e1,41
e4,5e3,5e2,5e1,5
...
n×n
=hn,2.
and
h1,21
h2,2h1,31
h3,2h2,3h1,41
h4,2h3,3h2,4h1,5
...
n×n
=en,n+1.
The cases a=k1 of Theorem 2.1 and k= 1 of Theorem 2.2 can be written
as
Corollary 2.4. For k, n > 0,
hn1
x1
,..., 1
xk+1 =det (eij+k(x1,...,xi+k))16i,j6n
(x1···xk+1)n
5
and
hk1
x1
,..., 1
xn+1 =det (hij+k(x1,...,xj+1))16i,j6n
(x1···xn+1)k.
We see that the identities (1.5) and (1.6) follow directly from this corollary,
taking into account (1.7), (1.8) and (2.1).
3 Proofs of Theorems
To prove the first theorem, we take into account that
ek,n =xnek1,n1+ek,n1.
We have
ea,k ea1,k ··· ean+1,k
ea+1,k+1 ea,k+1 ··· ean+2,k+1
.
.
..
.
..
.
.
ea+n2,k+n2ea+n3,k+n2··· ea1,k+n2
ea+n1,k+n1ea+n2,k+n1··· ea,k+n1
=
ea,k ea1,k ··· ean+1,k
ea+1,k+1 ea,k+1 ··· ean+2,k+1
.
.
..
.
..
.
.
ea+n2,k+n2ea+n3,k+n2··· ea1,k+n2
ea+n1,k+n2ea+n2,k+n2··· ea,k+n2
.
.
.
=
ea,k ea1,k ··· ean+1,k
ea+1,k ea,k ··· ean+2,k
.
.
..
.
..
.
.
ea+n2,k+n3ea+n3,k+n3··· ea1,k+n3
ea+n1,k+n2ea+n2,k+n2··· ea,k+n2
.
.
.
=
ea,k ea1,k ··· ean+1,k
ea+1,k ea,k ··· ean+2,k
.
.
..
.
..
.
.
ea+n2,k ea+n3,k ··· ea1,k
ea+n1,k ea+n2,k ··· ea,k
.
Theorem 2.1 is proved.
Considering the relation
hk,n =xnhk1,n +hk,n1,
6
we can write
ha,n+kha1,n+k··· han+2,n+khan+1,n+k
ha+1,n+kha,n+k··· han+3,n+khan+2,n+k
.
.
..
.
..
.
..
.
.
ha+n1,n+kha+n2,n+k··· ha+1,n+kha,n+k
=
ha,n1+kha1,n+k··· han+2,n+khan+1,n+k
ha+1,n1+kha,n+k··· han+3,n+khan+2,n+k
.
.
..
.
..
.
..
.
.
ha+n1,n1+kha+n2,n+k··· ha+1,n+kha,n+k
.
.
.
=
ha,n1+kha1,n1+k··· han+2,n1+khan+1,n+k
ha+1,n1+kha,n1+k··· han+3,n1+khan+2,n+k
.
.
..
.
..
.
..
.
.
ha+n1,n1+kha+n2,n1+k··· ha+1,n1+kha,n+k
.
.
.
=
ha,1+kha1,2+k··· han+2,n1+khan+1,n+k
ha+1,1+kha,2+k··· han+3,n1+khan+2,n+k
.
.
..
.
..
.
..
.
.
ha+n1,1+kha+n2,2+k··· ha+1,n1+kha,n+k
.
The proof of Theorem 2.2 is finished.
4 Proof of Corollary 2.4
We can write
hn1
x1
,..., 1
xk=s[n]1
x1
,..., 1
xk
=
(x1···xk)2ndet x(λj+kj)
i16i,j6k
(x1···xk)2ndet x(kj)
i16i,j6k
=
det x2n(λj+kj)
i16i,j6k
(x1···xk)ndet xn(kj)
i16i,j6k
=s[nk1](x1,...,xk)
(x1···xk)n,
where
λj=(n, j = 1,
0, j > 1.
7
According to Giambelli’s determinant formulas [1], we have
s[nk1](x1,...,xk) = det(eij+k1(x1,...,xk))16i,j6n.
Taking into account Theorem 2.1, the proof of the first relation is finished.
The proof of the second relation is similar to the proof of the first relation,
invoking another special case of Giambelli’s determinant formulas [1],
s[kn](x1,...,xn+1) = det(hij+k(x1,...,xn+1))16i,j 6n,
and Theorem 2.2.
5 Concluding remarks
The paper calculates some determinants involving the complete and elemen-
tary symmetric functions. Some specialization of these results are given in the
paper. These specializations are determinant formulas for Stirling numbers of
both kinds. The similar determinant formulas involving r-Stirling numbers,
r-Whitney numbers, Legendre-Stirling numbers, Jacobi-Stirling numbers, and
central factorial numbers can be obtained as well. This fact is possible be-
cause these numbers are specializations of complete and elementary symmetric
functions.
Acknowledgments. The author expresses his gratitude to Dr. Oana Merca
for the careful reading of the manuscript and helpful remarks. Finally, special
thanks go to the anonymous referee for many suggestions to and comments on
the original version of this paper.
References
[1] G. Z. Giambelli, Alcune propriet`a dele funzioni simmetriche caratteristiche,
Atti Torino,38, 823–844, 1903.
[2] I. G. Macdonald, Schur functions: Theme and variations, Publ. I.R.M.A.,
Strasbourg, Acte 28e, S´eminaire Lotharingien, 539, 1992.
[3] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Edition,
Clarendon Press, Oxford, 1995.
[4] M. Merca, A convolution for complete and elementary symmetric functions,
Aequationes Math. 86(3), 217–229, 2013.
[5] P. Mongelli, Combinatorial interpretations of particular evaluations of com-
plete and elementary symmetric functions, Electron. J. Combin.,19(1), 2012
#P60.
[6] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published
electronically at http://oeis.org, 2013.
8
... In the last few decades, the importance of symmetric polynomials has emerged in many branches of pure and applied mathematics, such as representation theory [1], algebraic combinatorics [2], and numerical analysis [3][4][5]. e symmetric polynomials have several types, for instance, monomial, complete, and the elementary, see [6][7][8][9] and references therein. roughout the current paper, we will focus on the elementary type. ...
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Alcune proprietà dele funzioni simmetriche caratteristiche
  • G Z Giambelli
  • GZ Giambelli
G. Z. Giambelli, Alcune proprietà dele funzioni simmetriche caratteristiche, Atti Torino, 38, 823–844, 1903.