Generalized Dyck tilings

Generalized Dyck tilings (Extended Abstract)

Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then w...

On Generalized Dyck Paths

We generalize the elegant bijective proof of the Chung Feller theorem from the paper of Young-Ming Chen [The Chung-Feller theorem revisited, Disc. Math. 308 (2008), 1328–1329].

Counting Generalized Dyck Paths

The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0, 0) to (n, n) which is below the diagonal line y = x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0, 0) to (m, n) ∈ N2 which is below the diagonal line y...

Dyck tilings , linear extensions , descents , and inversions ( extended abstract )

Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. 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 seco...

Generalized Tilings with Height