Jang Soo Kim (김장수), Proofs of Two Conjectures of Kenyon and Wilson on Dyck Tilings

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 talk 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.
This talk is based on the following paper: arxiv:1108.5558.