### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM NOVEMBER 20, 2009

**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.

**Time:** Friday, November 20, 2009, 3:30-4:20 p.m.
**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