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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

j16i,j6n

and B(k)

n=i+k

j16i,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

j16i,j6n

=k!n(1.1)

and

det i+k

j16i,j6n

=n!k.(1.2)

Experiment 2. Let

A(k)

n= i+k

j+k−116i,j6n

and B(k)

n= i+k

j+k−116i,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+k−116i,j6n

=n+k

k(1.3)

and

det i+k

j+k−116i,j6n

=n+k

k.(1.4)

Experiment 3. We consider the square matrices

A(k)

n=i+k

j+ 116i,j6n

and B(k)

n=i+k

j+ 116i,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 1−k!

jx−1

and

det i+k

j+ 116i,j6n

=k!n

k

X

j=1

(−1)j−11

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+ 116i,j6n

= (n+ 1)!k−1

n+1

X

j=1

(−1)j−11

jk−1n+ 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+n−j

i16i,j6n

det xn−j

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 }

k−n

).

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 (hi−j+1,n)16i,j6k

and

hk,n = det (ei−j+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(ei−j+a,i−1+k)16i,j6n= det(ei−j+a,k)16i,j6n.

Theorem 2.2. Let aand kbe two non-negative integers. Then

det(hi−j+a,j+k)16i,j6n= det(hi−j+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 (ei−j+k(x1, x2,...,xi−1+k))16i,j6n= (x1x2···xk)n.

By Theorem 2.2, with kreplaced by 0, we get

Corollary 2.2. For a, n > 0,

det (hi−j+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 (ei−j+1(x1, x2,...,xi−1+k))16i,j6n=hn(x1, x2,...,xk),and

det (hi−j+1(x1, x2,...,xj−1+k))16i,j6n=en(x1, x2,...,xn+k−1).

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=k−1 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 (ei−j+k(x1,...,xi+k))16i,j6n

(x1···xk+1)n

5

and

hk1

x1

,..., 1

xn+1 =det (hi−j+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 =xnek−1,n−1+ek,n−1.

We have

ea,k ea−1,k ··· ea−n+1,k

ea+1,k+1 ea,k+1 ··· ea−n+2,k+1

.

.

..

.

..

.

.

ea+n−2,k+n−2ea+n−3,k+n−2··· ea−1,k+n−2

ea+n−1,k+n−1ea+n−2,k+n−1··· ea,k+n−1

=

ea,k ea−1,k ··· ea−n+1,k

ea+1,k+1 ea,k+1 ··· ea−n+2,k+1

.

.

..

.

..

.

.

ea+n−2,k+n−2ea+n−3,k+n−2··· ea−1,k+n−2

ea+n−1,k+n−2ea+n−2,k+n−2··· ea,k+n−2

.

.

.

=

ea,k ea−1,k ··· ea−n+1,k

ea+1,k ea,k ··· ea−n+2,k

.

.

..

.

..

.

.

ea+n−2,k+n−3ea+n−3,k+n−3··· ea−1,k+n−3

ea+n−1,k+n−2ea+n−2,k+n−2··· ea,k+n−2

.

.

.

=

ea,k ea−1,k ··· ea−n+1,k

ea+1,k ea,k ··· ea−n+2,k

.

.

..

.

..

.

.

ea+n−2,k ea+n−3,k ··· ea−1,k

ea+n−1,k ea+n−2,k ··· ea,k

.

Theorem 2.1 is proved.

Considering the relation

hk,n =xnhk−1,n +hk,n−1,

6

we can write

ha,n+kha−1,n+k··· ha−n+2,n+kha−n+1,n+k

ha+1,n+kha,n+k··· ha−n+3,n+kha−n+2,n+k

.

.

..

.

..

.

..

.

.

ha+n−1,n+kha+n−2,n+k··· ha+1,n+kha,n+k

=

ha,n−1+kha−1,n+k··· ha−n+2,n+kha−n+1,n+k

ha+1,n−1+kha,n+k··· ha−n+3,n+kha−n+2,n+k

.

.

..

.

..

.

..

.

.

ha+n−1,n−1+kha+n−2,n+k··· ha+1,n+kha,n+k

.

.

.

=

ha,n−1+kha−1,n−1+k··· ha−n+2,n−1+kha−n+1,n+k

ha+1,n−1+kha,n−1+k··· ha−n+3,n−1+kha−n+2,n+k

.

.

..

.

..

.

..

.

.

ha+n−1,n−1+kha+n−2,n−1+k··· ha+1,n−1+kha,n+k

.

.

.

=

ha,1+kha−1,2+k··· ha−n+2,n−1+kha−n+1,n+k

ha+1,1+kha,2+k··· ha−n+3,n−1+kha−n+2,n+k

.

.

..

.

..

.

..

.

.

ha+n−1,1+kha+n−2,2+k··· ha+1,n−1+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+k−j)

i16i,j6k

(x1···xk)2ndet x−(k−j)

i16i,j6k

=

det x2n−(λj+k−j)

i16i,j6k

(x1···xk)ndet xn−(k−j)

i16i,j6k

=s[nk−1](x1,...,xk)

(x1···xk)n,

where

λj=(n, j = 1,

0, j > 1.

7

According to Giambelli’s determinant formulas [1], we have

s[nk−1](x1,...,xk) = det(ei−j+k−1(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(hi−j+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