Dyck tilings, increasing trees, descents, and inversions

Abstract

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.

24 pages, 9 figures

References in corpus (3)

• Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
• Fully Packed Loops in a triangle: matchings, paths and puzzles
• Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Cited by in corpus (9)

• Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs
• Boundary correlations in planar LERW and UST
• Ballot tilings and increasing trees
• Classical and consecutive pattern avoidance in rooted forests
• Pfaffian formulas for spanning tree probabilities
• Bijections on Dyck tilings: DTS/DTR bijections, Dyck tableaux and tree-like tableaux
• Symmetric Dyck tilings, ballot tableaux and tree-like tableaux of shifted shapes
• The $m$-Cover Posets and Their Applications
• Dyck tilings of type $D$