Lattices of Paths: Representation Theory and Valuations

TL;DR

This paper explores distributive lattices derived from lattice path combinatorics, analyzing their spectrum and explicitly computing the Euler characteristic based on natural path parameters.

Contribution

It provides a detailed description of the spectrum and explicit Euler characteristic formulas for Dyck, Motzkin, and Schroder lattices, linking combinatorics and lattice theory.

Findings

Spectrum characterization of the lattices

Explicit Euler characteristic formulas

Connections between lattice paths and distributive lattices

Abstract

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schroder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths.