### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM

AUGUST 29, 2014

**Speaker: **
Matt Beck, San Francisco State University

**Title: ***
Combinatorial Reciprocity Theorems
*

**Abstract:**
A common theme of enumerative combinatorics is formed by counting functions
that are polynomials.
For example, one proves in any introductory graph theory course
that the number of proper k-colorings of a given graph G is a polynomial in k, the
chromatic polynomial of G. Combinatorics is abundant with polynomials that count
something when evaluated at positive integers, and many of these polynomials have
a (completely different) interpretation when evaluated at negative integers: these
instances go by the name of combinatorial reciprocity theorems. For example, when
we evaluate the chromatic polynomial of G at -1, we obtain (up to a sign) the number
of acyclic orientations of G, that is, those orientations of G that do not contain
a coherently oriented cycle.

Reciprocity theorems appear all over combinatorics. This talk will attempt to show
some of the charm (and usefulness!) these theorems exhibit. Our goal is to weave
a unifying thread through various combinatorial reciprocity theorems, by looking
at them through the lens of geometry.

**Time:** Friday, August 29, 2014, 3:30-4:20 p.m.

**Place:** Exploratory Hall, room 4106

**Refreshments** will be served at 3:00 p.m.

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