Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Kim, Jang Soo

关键词: Dyck paths;    Dyck tilings;    Hermite histories;    Matchings;    Hermite polynomials;   

DOI  :  10.1016/j.jcta.2012.05.008

来源: Elsevier

【 摘 要 】

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. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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