Dyck tilings, increasing trees, descents, and inversions

TL;DR

This paper explores the combinatorial structure of cover-inclusive Dyck tilings, establishing bijections with linear extensions of tree posets and connecting tiling statistics to inversions and descents.

Contribution

It introduces two bijections linking Dyck tilings to linear extensions, revealing new combinatorial interpretations of tiling statistics.

Findings

Bijection between (area + tiles)/2 and inversions in linear extensions

Bijection between boundary discrepancy and descents in linear extensions

Connections to statistical physics models and Kazhdan--Lusztig polynomials

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.