Summary: Alternating sign matrices and tilings of
David E. Anderson
March 31, 2002
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-
Gessel-Viennot theorem to express the number in terms of determinants.
1 Introduction 1