250121 VO Topics in Combinatorics (2022W)

Enumeration of Tilings

The enumeration of tilings of regions by given tiles (such as dominoes, rhombi, etc.) may have its origin in recreational mathematics, but has been intensively studied since at least 50 years in combinatorics and statistical physics (there, in the guise of the dimer model) alike. The prototypical example of a tiling enumeration problem is the problem of finding the number of ways to cover an m x n chessboard completely by dominoes (2x1 tiles). Given that one of m or n is even, it turns out that this number is given by a closed formula that is somewhat surprising in its complexity.

The goal of this course is to provide an introduction into this fascinating area, mainly from the point of view of Enumerative Combinatorics. (At the end, I may also briefly touch upon - equally fascinating - probabilistic aspects.) We shall encounter various techniques to enumerate tilings, as for example condensation, non-intersecting lattice paths, Kasteleyn determinants, Ciucu's matchings factorization theorem. Since determinants play a predominant role here, we shall also learn a few tricks how to evaluate determinants.
Guessing formulas for the terms of a sequence (a(n)) from its first few values will also be discussed.

It will not be necessary to have already attended the "Combinatorics" course.