Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

TL;DR

This paper proves two conjectures by Kenyon and Wilson relating the inverse of a matrix to Dyck tilings, revealing new combinatorial identities and a bijection with complete matchings.

Contribution

The paper proves two conjectures connecting matrix entries to Dyck tilings and establishes a bijection with complete matchings, advancing combinatorial understanding.

Findings

Sum of absolute values of each row/column of M^{-1} equals the number of Dyck tilings.

Total sum of absolute values of M^{-1} entries equals the number of complete matchings.

Established a bijection between Dyck tilings and complete matchings.

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.