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Dyck tilings, increasing trees, descents, and inversions

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arxiv 1205.6578 v4 pith:RNPYTCP7 submitted 2012-05-30 math.CO

Jang Soo Kim , Karola Meszaros , Greta Panova , David B. Wilson This is my paper · ORCID
classification math.CO
keywords tilingsdycktileslinearbijectioncover-inclusivedescentsextension
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Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and also Kazhdan--Lusztig polynomials. We give two bijections between cover-inclusive Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy" between the upper and lower boundary of the tiling to descents of the linear extension.

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  1. Jones--Wenzl projections of type $D$ and Dyck tilings

    math.CO 2024-12 conditional novelty 6.0 of 10

    Coefficients of the type D Jones-Wenzl projection are expressed as generating functions over Dyck tilings decorated with bi-colored vertical Hermite histories, for both even and odd dot cases.

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