Pfaffian Algorithms for Sampling Routings on Regions with Free Boundary Conditions

Sets of non-intersecting, monotonic lattice paths, or xed routings, provide a common representation for several combinatorial problems and have been the key element for designing sampling algorithms. Markov chain algorithms based on routings have led to eecient samplers for tilings, Eulerian orientations 8] and triangulations 9], while an algorithm which successively calculates ratios of determinants has led to a very fast method for sampling xed routings 12]. We extend Wil-son's determinant algorithm 12] to sample free routings where the number of paths, as well as the endpoints, are allowed to vary. The algorithm is based on a technique due to Stembridge for counting free routings by calculating the Pfaaan of a suitable matrix 11] and a method of Colbourn, Myrvold and Neufeld 1] for eeciently calculating ratios of determinants. As an application, we show how to sample tilings on pla-nar lattice regions with free boundary conditions.