An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are h-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic a...

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W , such that S (but not necessarily R) is reduced. For each such pair (R, S) we construct a family of W -invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters q, t1, t2, . . . , tr, where r (= 1, 2 or 3) is the number of W -orbits in...

A geometric perspective is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and tdesigns. Index Terms – Frames, Grammian, Homogeneous polynomials, Symmetric tensors, t-designs, Welch bounds

We present several new and compact formulas for the modified integral form of Macdonald polynomials, building on “multiline queue” formula polynomials due to Corteel, Mandelshtam, Williams. also introduce a quasisymmetric analogue polynomials. These “quasisymmetric polynomials" refine (symmetric) specialize Schur defined by Haglund, Luoto, Mason, van Willigenburg.

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...

Macdonald defined in [M1] a remarkable class of symmetric polynomials Pλ(x; q, t) which depend on two parameters and interpolate between many families of classical symmetric polynomials. For example Pλ(x; t) = Pλ(x; 0, t) are the Hall-Littlewood polynomials which themselves specialize for t = 0 to Schur functions sλ. Also Jack polynomials arise by taking q = t and letting t tend to 1. The Hall-...

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities, and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themse...

We introduce a general class of symmetric polynomials that have saturated Newton polytope and their has integer decomposition property. The covers numerous previously studied polynomials.

Vector-valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik–Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r, p,N) and studi...