Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Authors:
Jang Soo Kim
Affiliations:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2012

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Abstract

Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix M^-^1 is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of M^-^1. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of M^-^1 is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.