Rational Dyck paths and decompositions

TL;DR

This paper explores the combinatorial structure of rational Dyck paths by decomposing them into tuples of simpler Dyck paths, establishing new correspondences and dualities with applications to binary trees.

Contribution

It introduces a novel decomposition of rational Dyck paths into tuples and relates this to various combinatorial models, providing new insights into their orderings and dualities.

Findings

Decomposition of rational Dyck paths into tuples of Dyck paths.

Reinterpretation of Young and rotation orders via decomposition.

Duality between (a,b)-Dyck and (b,a)-Dyck paths using binary trees.

Abstract

We study combinatorial properties of a rational Dyck path by decomposing it into a tuple of Dyck paths. The combinatorial models such as $b$-Stirling permutations, $(b+1)$-ary trees, parenthesis presentations, and binary trees play central roles to establish a correspondence between the rational Dyck path and the tuple of Dyck paths. We reinterpret two orders, the Young and the rotation orders, on rational Dyck paths in terms of the tuple of Dyck paths by use of the decomposition. As an application, we show a duality between $(a,b)$-Dyck paths and $(b,a)$-Dyck paths in terms of binary trees.