scholarly journals Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

10.37236/1044 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Andrius Kulikauskas ◽  
Jeffrey Remmel
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Transition Matrices ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations ◽  
Lyndon Words

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions

1959 ◽  
Vol 11 ◽  
pp. 383-396 ◽  
Author(s):  
Marvin Marcus ◽  
Roger Purves
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Similarity Transformation ◽  
Elementary Symmetric Function ◽  
Linear Transformations ◽  
Elementary Symmetric Functions ◽  
Image Position

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

An Inequality for Elementary Symmetric Functions

1972 ◽  
Vol 15 (1) ◽  
pp. 133-135 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Geometric Mean ◽  
Binomial Coefficient ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Image Position

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

scholarly journals The asymptotic behavior of elementary symmetric functions on a probability distribution

2001 ◽  
Vol 14 (3) ◽  
pp. 237-248 ◽  
Author(s):  
V. S. Kozyakin ◽  
A. V. Pokrovskii
Keyword(s):  
Asymptotic Behavior ◽  
Probability Distribution ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Random Mappings

The problem on asymptotic of the value π(m,n)=m!σm(p(1,n),p(2,n),…,p(n,n)) is considered, where σm(x1,x2,…,xn) is the mth elementary symmetric function of n variables. The result is interpreted in the context of nonequiprobable random mappings theory.

Probability and the elementary symmetric functions

1973 ◽  
Vol 74 (1) ◽  
pp. 133-139 ◽  
Author(s):  
J. Denmead Smith
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Random Variables ◽  
Elementary Symmetric Function ◽  
Independent Random Variables ◽  
Elementary Symmetric Functions ◽  
Congruent Modulo

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.

Inequalities for Generalized Symmetric Functions

1969 ◽  
Vol 12 (5) ◽  
pp. 615-623 ◽  
Author(s):  
K.V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Generating Series ◽  
Image Position ◽  
Complete Symmetric Function

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.

scholarly journals Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions

10.37236/2131 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Pietro Mongelli
Keyword(s):  
Spectral Theory ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial  interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.

scholarly journals On the expression of a symmetric function in terms of the elementary symmetric functions

1888 ◽  
Vol 7 ◽  
pp. 41-42
Author(s):  
R. E. Allardice
Keyword(s):  
Rational Function ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Proof ◽  
Elementary Symmetric Functions ◽  
Definition Of

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.

scholarly journals On Determinants of Symmetric Functions

1927 ◽  
Vol 1 (1) ◽  
pp. 55-61 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
The Other ◽  
Elementary Symmetric Functions ◽  
The Difference ◽  
Image Position

The result of dividing the alternant |aαbβcγ…| by the simplest alternant |a0b1c2…| (the difference-product of a, b, c, …) is known to be a symmetric function expressible in two distinct ways, (1) as a determinant having for elements the elementary symmetric functions C, of a, b, c, …, (2) as a determinant having for elements the complete homogeneous symmetric functions Hr. For exampleThe formation of the (historically earlier) H-determinant is evident. The suffixes in the first row are the indices of the alternant; those of the other rows decrease by unit steps. This result is due to Jacobi.

scholarly journals Schur and $e$-Positivity of Trees and Cut Vertices

10.37236/8930 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Samantha Dahlberg ◽  
Adrian She ◽  
Stephanie Van Willigenburg
Keyword(s):  
Linear Combination ◽  
Connected Graph ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Perfect Matching ◽  
Schur Functions ◽  
Connected Components ◽  
Elementary Symmetric Functions ◽  
Chromatic Symmetric Function ◽  
Positive Linear Combination

We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geqslant \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geqslant\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.

scholarly journals A note on extrema of linear combinations of elementary symmetric functions

2012 ◽  
Vol 60 (2) ◽  
pp. 219-224 ◽  
Author(s):  
Alexander Kovačec ◽  
Salma Kuhlmann ◽  
Cordian Riener
Keyword(s):  
Symmetric Functions ◽  
Linear Combinations ◽  
Elementary Symmetric Functions
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