Intervals in Dyck paths and the wreath conjecture

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 ${\iota }_k(k,l)={\left(\genfrac{}{}{0px}{}{k}{l}\right)}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.