Terms Of Elementary Symmetric Functions Research Articles

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

We give a new characterization of the vertical-strip LLT polynomials GP(x;q) as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if q is replaced by q+1, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

De Boor–Fix functionals and Hermite boundary conditions in the polynomial spline interpolation problem

By means of de Boor–Fix functionals Hermite boundary conditions in the problem of spline interpolation are obtained. It is shown that some of the first and last B-spline coefficients can be found by explicit formulas in terms of elementary symmetric functions and the remaining coefficients can be computed as a solution of a banded system of linear equations.

Natural generalization of the ground-state Slater determinant to more than one dimension

The basic question is addressed, how the space dimension $d$ is encoded in the Hilbert space of $N$ identical fermions. There appears a finite number $N!^{d-1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonised into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the $N$-fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.

Bijections on [formula omitted]-level rook placements

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. This bijection also preserves the m-inversion number statistic of an m-level rook placement, defined by Briggs and Remmel. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards, but it yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne.

Bijections on m-level Rook Placements

Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne. Nous considérons les rangs d’un échiquier partagés en ensembles de $m$ rangs appelés les niveaux. Un $m$-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de $m$-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de $m$-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne.

Double point curves for corank 2 map germs from [formula omitted] to [formula omitted

We characterize finite determinacy of map germs f : ( C 2 , 0 ) → ( C 3 , 0 ) in terms of the Milnor number μ ( D ( f ) ) of the double point curve D ( f ) in ( C 2 , 0 ) and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs f t : ( C 2 , 0 ) → ( C 3 , 0 ) is equivalent to the constancy of both μ ( D ( f t ) ) and μ ( f t ( C 2 ) ∩ H ) with respect to t, where H ⊂ C 3 is a generic plane.

Expansion and linear complexity of the coordinate sequences over Galois rings

The coordinate sequences of the trace sequences over a Galois ring defined by the trace function are used significantly in cryptography, coding and communication applications. In this paper, a p -adic expansion for the coordinate sequences in terms of elementary symmetric functions is provided for the case that the characteristic p of the residue field of the Galois ring is an arbitrary prime, which generalizes the related result of Kumar and Helleseth for the characteristic being p = 2 . From the expression, upper and lower bounds on the linear complexity of the coordinate sequences are derived.

Counting Lattice Points of Rational Polyhedra

The generating function F(P)=∑α∈P∩ZNxα for a rational polytope P carries all essential information of P. In this paper we show that for any positive integer n, the generating function F(P, n) of nP={nx:x∈P} can be written as F(P, n)=∑α∈APα(n)xnα, where A is the set of all vertices of P and each Pα(n) is a certain periodic function of n. The Ehrhart reciprocity law follows automatically from the above formula. We also present a formula for the coefficients of Ehrhart polynomials in terms of elementary symmetric functions.

Binary Sequences of Period 2m−1 with Large Linear Complexity

The base-2 expansion of trace sequences over Galois rings of characteristic 2 e and size 2 em gives rise to e binary coordinate sequences of increasing linear complexity. An expression for the coordinate sequences in terms of elementary symmetric functions is provided. For a class of the coordinate sequences derived from the base-2 expansion of the trace sequences of characteristic 2 3 , the linear complexity is determined exactly.

An Expansion for the Coordinates of the Trace Function over Galois Rings

The base-2 expansion of the trace sequence over Galois rings of characteristic 2k and size 2km gives rise to k binary coordinate sequences of increasing linear span. An expression for these sequences in terms of elementary symmetric functions is provided. As a byproduct, bounds on the linear span of the coordinate sequences are easily derived.

Explicit formula for the multiple generating function of product of generalized Laguerre polynomials

Using an operator approach, Messina and Paladimo have recently demonstrated a method of evaluating a multiple generating function of the product of generalized Laguerre polynomials, and arrived at the solution which is given implicitly. By using an addition theorem of Vilenkin for the Laguerre polynomials, which has its origin in group theory, we are able to give an explicit formula for the multiple generating function in terms of elementary symmetric functions. It is further shown that the multiple generating function is a symmetric function, an interesting property which is not altogether obvious from the solution via the operator approach.

A Generalization of Waring's Formula

Waring's formula for expressing power sum symmetric functions in terms of elementary symmetric functions is generalized to monomial symmetric functions with equal exponents by applying the orthogonality property of Ramanujan sums together with the Möbius function over the set partition lattice.

A Symmetric Function Generalization of the Chromatic Polynomial of a Graph

For a finite graph G with d vertices we define a homogeneous symmetric function XG of degree d in the variables x1, x2, ... . If we set x1 = ... = xn= 1 and all other xi = 0, then we obtain χG(n), the chromatic polynomial of G evaluated at n. We consider the expansion of XG in terms of various symmetric function bases. The coefficients in these expansions are related to partitions of the vertices into stable subsets, the Möbius function of the lattice of contractions of G, and the structure of the acyclic orientations of G. The coefficients which arise when XG is expanded in terms of elementary symmetric functions are particularly interesting, and for certain graphs are related to the theory of Hecke algebras and Kazhdan-Lusztig polynomials.

Computing elliptic integrals by duplication

Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) forR-polynomials, as well as an upper bound for the error made in truncating the Taylor series of anR-function.