Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks

TL;DR

This paper develops bijections to Motzkin paths and quarter-plane walks to enumerate interval-closed sets in specific posets, providing new generating functions and functional equations for these combinatorial structures.

Contribution

It introduces novel bijections linking interval-closed sets to Motzkin paths and quarter-plane walks, enabling enumeration of these sets in various posets.

Findings

Generated explicit formulas for interval-closed sets in product of chains.

Derived functional equations for truncated rectangle posets.

Connected combinatorial structures to known path models.

Abstract

We find a generating function for interval-closed sets of the product of two chains poset by constructing a bijection to certain bicolored Motzkin paths. We also find a functional equation for the generating function of interval-closed sets of truncated rectangle posets, including the type $A$ root poset, by constructing a bijection to certain quarter-plane walks.