 
Summary: Alternating sign matrices and tilings of
Aztec rectangles
Undergraduate Thesis
David E. Anderson
Columbia University
anderson@math.columbia.edu
March 31, 2002
Abstract
The problem of counting numbers of tilings of certain regions has long interested re
searchers in a variety of disciplines. In recent years, many beautiful results have been
obtained related to the enumeration of tilings of particular regions called Aztec diamonds.
Problems currently under investigation include counting the tilings of related regions with
holes and describing the behavior of random tilings.
Here we derive a recurrence relation for the number of domino tilings of Aztec rectan
gles with squares removed along one or both of the long edges. Through an interpretation
of a sequence of alternating sign matrix rows as a family of nonintersecting lattice paths,
we relate this enumeration to that of lozenge tilings of trapezoids, and use the Lindstršom
GesselViennot theorem to express the number in terms of determinants.
Contents
1 Introduction 1
