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

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.

21 pages, 11 figures, one table