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 ^-11 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 ^-11. 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.