The enumeration of generalized Tamari intervals

Let v be a grid path made of north and east steps. The lattice Tam(v), based on all grid paths weakly above v and sharing the same endpoints as v , was introduced by Preville-Ratelle and Viennot (2016) and corresponds to the usual Tamari lattice in the case v = ( N E ) n . Our main contribution is that the enumeration of intervals in Tam(v), over all v of length n , is given by 2 ( 3 n + 3 ) ! ( n + 2 ) ! ( 2 n + 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.

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