Enumeration of tilings of diamonds and hexagons with defects
Harald Helfgott, Ira M. Gessel
TL;DR
This paper develops determinant-based methods to count tilings of Aztec diamonds and hexagons with defects, providing closed-form solutions for several enumeration problems in combinatorics.
Contribution
It introduces a determinant approach to enumerate tilings with defects, solving multiple open problems in the enumeration of matchings.
Findings
Determinant formulas for tilings with defects
Closed-form solutions for specific enumeration problems
Extension of tiling enumeration to defective regions
Abstract
We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Mathematical Dynamics and Fractals · Markov Chains and Monte Carlo Methods