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Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbontiles shaped like Dyck paths, in which tiles are no larger than the tiles theycover. These tilings arise in the study of certain statistical physics modelsand also Kazhdan--Lusztig polynomials. We give two bijections betweencover-inclusive Dyck tilings and linear extensions of tree posets. The firstbijection maps the statistic (area + tiles)/2 to inversions of the linearextension, and the second bijection maps the "discrepancy" between the upperand lower boundary of the tiling to descents of the linear extension.

Dyck tilings, increasing trees, descents, and inversions