Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers

Authors:
Richard A. Brualdi;Stephen Kirkland
Affiliations:
Department of Mathematics, University of Wisconsin, Madison, 480 Lincoln Drive, Madison, WI;Department of Mathematics, University of Regina, Regina, Sask., Canada S45 0A2
Venue:
Journal of Combinatorial Theory Series B
Year:
2005

Citing 5
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Abstract

The Aztec diamond of order n is a certain configuration of 2n(n + 1) unit squares. We give a new proof of the fact that the number Πn of tilings of the Aztec diamond of order n with dominoes equals 2n(n+1)/2. We determine a sign-nonsingular matrix of order n (n + 1) whose determinant gives Πn. We reduce the calculation of this determinant to that of a Hankel matrix of order n whose entries are large Schröder numbers. To calculate that determinant we make use of the J-fraction expansion of the generating function of the Schröder numbers.