The number of subgroups of type n and cotype v in a finite abelian p-group of type A is a polynomial g$v(p) with integral coefficients. We prove g u v ( p ) has nonnegative coefficients for all partitions p and v if and only if no two parts of A differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certa...

We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary symmetric polynomial or a complete homogeneous symmetric polynomial. Thus, we generalize the classical Pieri’s rule for symmetric polynomials/Grassmann varietie...

The profile of a relational structure R is the function φR which counts for every integer n the number φR(n), possibly infinite, of substructures of R induced on the n-element subsets, isomorphic substructures being identified. If φR takes only finite values, this is the Hilbert function of a graded algebra associated with R, the age algebra KA(R), introduced by P. J. Cameron. In this paper we ...

Motivated by certain conjectures regarding immanants of Jacobi-Trudi matrices, Stanley has recently defined and studied a symmetric function generalization XG of the chromatic polynomial of a graph G. Independently, Chung and Graham have defined and studied a directed graph invariant called the cover polynomial. The cover polynomial is closely related to the chromatic polynomial and to the rook...

Standard algorithms for dealing with symmetric polynomials are presented using rewriting techniques. This is for instance the case of the ”fundamental theorem of symmetric polynomials”, which states that any symmetric polynomial is a polynomial in the elementary symmetric ones, and whose proof involves rewriting using a suitable elimination order. This kind approach usually spoils useful featur...

We give two examples where symmetric polynomials play an important rôle in physics: First, the partition functions of ideal quantum gases are closely related to certain symmetric polynomials, and a part of the corresponding theory has a thermodynamical interpretation. Further, the same symmetric polynomials also occur in Berezin’s theory of quantization of phase spaces with constant curvature.

We investigate the validity of the bootstrap method for the elementary symmetric polynomials S both xed and increasing order k, as n ! 1 the cases where = EX 1 6 = 0, the nondegenerate case, and where = EX 1 = 0, the degenerate case, are considered.

Given g1, . . . , gs ∈ R[X] = R[X1, . . . , Xn] such that the semialgebraic set K := {x ∈ R | gi(x) ≥ 0 for all i} is compact. Schmüdgen’s Theorem says that if f ∈ R[X] such that f > 0 on K, then f is in the preordering in R[X] generated by the gi’s, i.e., f can be written as a finite sum of elements σg1 1 . . . g es s , where σ is a sum of squares in R[X] and each ei ∈ {0, 1}. Putinar’s Theore...

We study the problem of representing symmetric Boolean functions as symmetric polynomials over . We show an equivalence between such representations and simultaneous communication protocols. Computing a function with a polynomial of degree modulo is equivalent to a two player simultaneous protocol for computing where one player is given the first digits of the weight in base and the other is gi...