The enumeration of generalized Tamari intervals

Let vv be a grid path made of north and east steps. The lattice Tam(v), based on all grid paths weakly above vv and sharing the same endpoints as vv, was introduced by Préville-Ratelle and Viennot (2016) and corresponds to the usual Tamari lattice in the case v=(NE)nv=(NE)n. Our main contribution is that the enumeration of intervals in Tam(v), over all vv of length nn, is given by 2(3n+3)!(n+2)!(2n+3)!. This formula was first obtained by Tutte (1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in Tam(v) to non-separable planar maps.