Intervals in Dyck paths and the wreath conjecture

TL;DR

This paper derives a formula for counting specific intervals in Dyck paths and explores related conjectures, providing new combinatorial insights and proposing stronger variants of the wreath conjecture.

Contribution

It introduces a formula for intervals in Dyck paths with fixed falls and proposes stronger variants of the wreath conjecture based on these findings.

Findings

Derived a formula for _{k}(m,l) and showed _{k}(k,l)=inom{k}{l}^2

Connected interval counts in Dyck paths to the wreath conjecture

Proposed stronger variants of the wreath conjecture for n=2k+1

Abstract

Let $\iota_{k}(m,l)$ denote the total number of intervals of length $m$ across all Dyck paths of semilength $k$ such that each interval contains precisely $l$ falls. We give the formula for $\iota_{k}(m,l)$ and show that $\iota_{k}(k,l)=\binom{k}{l}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.