Intervals in Dyck paths and the wreath conjecture

Abstract

Let ${\hat{I}}_k^1(m,l)$ 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 ${\hat{I}}_k^1(m,l)$ and show that ${\hat{I}}_k^1(k,l)={\left(\genfrac{}{}{0ex}{}{k}{l}\right)}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.

9 pages, 5 figures; corrected typos