## Seminar:

Fall 2008, Thursdays, Milner 216, 4:00-4:50 PM
__ Date:__ ** September 18 **

**" Hyperpfaffians in Combinatorics "**

**Daniel Redelmeier**

__Abstract__ In this talk we will discuss the hyperpfaffian, which is an extension of the pfaffian, which is in turn a generalisation of the determinant. We will spend the first part of the talk examining the different definitions used for the hyperpfaffian, as there are several non-equivalent forms. Then we will examine uses of the hyperpfaffian in combinatorics. The first is the Hyperpfaffian-Cactus theorem, which is related to the classical matrix-tree theorem and the later pfaffian-tree theorem. Second we will look at hyperpfaffian orientations, which can be used to count perfect matchings on a hyperpfaffian. If time permits we will look at hyperpfaffian rings/ideals, and specifically look at the fact that unlike the pfaffian ideal, the hyperpfaffian ideal is not an algebra with straightening law (this will also be explained in the talk).

This talk should be fairly self contained, but some knowledge of determinants, permutations and their sign, matchings, rings, and ideals would be helpful. Certainly knowledge of the hyperpaffian or even the pfaffian is not expected.