Speaker: Gabor Hetyei, University of North Carolina at Charlotte

Title: Geometric interpretations of the relation between Delannoy numbers and Legendre polynomials

Abstract: It has been known for over half a century that the central Delannoy numbers may be obtained by substituting three into the Legendre polynomials, but the relation was mostly dismissed as a "coincidence". Being convinced that there is no coincidence in mathematics, we will discuss up to two geometric interpretations of this relation.

Our focus will be on an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x-1)/2 in the n-th Legendre polynomial. We will see that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open generalized orthant. The polytopes we construct are closely related to the root polytopes introduced by Gelfand, Graev, and Postnikov.

Time permitting we will also discuss a join operation on colored simplicial complexes that preserves the Cohen-Macaulay property. Using this operation, the connection between Legendre polynomials and Delannoy numbers may be put in a wider context in a completely different way.

Place: Science and Technology Building I, Room 242

Refreshments will be served before the talk at 3:00 p.m. in Room 222.

Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491