  We count the number of linear intervals in the Tamari and the Dyck lattices
according to their height, using generating series and Lagrange inversion.
Surprisingly, these numbers are the same in both lattices. We define a new
family of posets on Dyck paths, which we call alt-Tamari posets. Each
alt-Tamari poset depends on the choice of an increment function delta in
{0,1}^n. We recover the Tamari and the Dyck lattices as extreme cases with
delta = 1 and delta = 0, respectively. We prove that all the alt-Tamari posets
have the same number of linear intervals of any given height.
