Jang Soo Kim
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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)
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https://doi.org/10.46298/dmtcs.3046
Proofs of two conjectures of Kenyon and Wilson on Dyck tilingsArticle
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.
Richard W. Kenyon;David Bruce Wilson, 2017, The Space of Circular Planar Electrical Networks, arXiv (Cornell University), 31, 1, pp. 1-28, 10.1137/140997798, https://arxiv.org/abs/1411.7425.
Jang Soo Kim;Karola Mészáros;Greta Panova;David B. Wilson, 2014, Dyck tilings, increasing trees, descents, and inversions, arXiv (Cornell University), 122, pp. 9-27, 10.1016/j.jcta.2013.09.008, https://arxiv.org/abs/1205.6578.
Richard W. Kenyon;David B. Wilson, 2014, Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs, arXiv (Cornell University), 28, 4, pp. 985-1030, 10.1090/s0894-0347-2014-00819-5, https://arxiv.org/abs/1107.3377.