Enumeration of labeled trees and Dyck tilings

TL;DR

This paper explores the combinatorial structure of labeled trees and Dyck tilings, providing factorization formulas, dualities, and decompositions that deepen understanding of their poset relationships and enumerations.

Contribution

It introduces a new cover relation on rational Dyck tilings, establishes duality between increasing and decreasing labelings, and presents two decompositions of rational Dyck tilings.

Findings

Factorization of generating functions for certain posets.

Duality between increasing and decreasing labelings.

Decomposition of rational Dyck tilings into simpler components.

Abstract

We study a partially ordered set of planar labeled rooted trees by use of combinatorial objects called Dyck tilings. A generating function of the poset is factorized when the minimum element of the poset is $312$-avoiding and satisfies some extra condition. We define a cover relation on rational Dyck tilings by that of labeled trees, and show that increasing and decreasing labelings are dual to each other. We consider two decompositions of a rational $(a,b)$-Dyck tiling: one is into $ab$ Dyck tilings and the other is into $a$ $(1,b)$-Dyck tilings. In the first case, we show that the weight of the $(a,b)$-Dyck tiling is the sum of the weights of $ab$ Dyck tilings. In the second case, we introduce a cover relation on $(1,b)$-Dyck tilings and obtain a poset of $(a,b)$-Dyck tilings by this decomposition.