The Ascent Lattice on Dyck Paths

In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevière, where cover relations replace a factor $DU^k D$ by $U^kD^2$. By relating this poset to another poset recently defined by Nadeau and Tewari, we prove that this still yields a lattice, which we call the ascent lattice $\mathbb D_n$. We then count intervals in $\mathbb D_n$. Their generating function is found to be algebraic of degree $3$. The proof is based on a recursive decomposition of intervals involving two catalytic parameters. The solution of the corresponding functional equation is inspired by recent work on the enumeration of walks confined to a quadrant. We also consider the order induced in $\mathbb D_{mn}$ on $m$-Dyck paths, that is, paths in which all ascent lengths are multiples of $m$, and on mirrored $m$-Dyck paths, in which all descent lengths are multiples of $m$. The first poset $\mathbb D_{m,n}$ is still a lattice for any $m$, while the second poset $\mathbb D'_{m,n}$ is only a join semilattice when $m>1$. In both cases, the enumeration of intervals is still described by an equation in two catalytic variables. Interesting connections arise with the sylvester congruence of Hivert, Novelli and Thibon, and again with walks confined to a quadrant. We combine the latter connection with probabilistic results to give asymptotic estimates of the number of intervals in both $\mathbb D_{m,n}$ and $\mathbb D'_{m,n}$. Their form implies that the generating functions of intervals are no longer algebraic, nor even D-finite, when $m>1$.