Unimodality and Dyck paths

TL;DR

This paper introduces a novel recursive approach to studying the rank-unimodality of Dyck lattices, using ECO methodology, and explores related polynomial families and succession rules.

Contribution

It presents a new recursive construction for Dyck paths, proposes polynomial analogs of ballot numbers, and initiates a systematic study of succession rules for unimodality.

Findings

A family of polynomials related to Dyck paths is described.

A succession rule is proposed to analyze unimodality.

Initial investigations into the unimodality of succession rules are presented.

Abstract

We propose an original approach to the problem of rankunimodality for Dyck lattices. It is based on a well known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. Even if we are not able to prove that Dyck lattices are rank-unimodal, we describe a family of polynomials (which constitutes a polynomial analog of ballot numbers) and a succession rule which appear to be useful in addressing such a problem. At the end of the paper, we also propose and begin a systematic investigation of the problem of unimodality of succession rules.