Jang Soo Kim - Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

dmtcs:3046 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) - https://doi.org/10.46298/dmtcs.3046
Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Authors: Jang Soo Kim

    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.


    Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
    Section: Proceedings
    Published on: January 1, 2012
    Imported on: January 31, 2017
    Keywords: Dyck paths, Dyck tilings, matchings, Hermite histories, orthogonal polynomials,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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    Source : ScholeXplorer IsReferencedBy DOI 10.48550/arxiv.1702.03261
    Source : ScholeXplorer IsReferencedBy DOI 10.1007/s00220-019-03615-0
    Source : ScholeXplorer IsReferencedBy ARXIV 1702.03261
    • 10.48550/arxiv.1702.03261
    • 10.1007/s00220-019-03615-0
    • 10.1007/s00220-019-03615-0
    • 10.1007/s00220-019-03615-0
    • 1702.03261
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