A recursion on maximal chains in the Tamari lattices

The Tamari lattices have been intensely studied since their introduction by Dov Tamari around 1960. However oddly enough, a formula for the number of maximal chains is still unknown. This is due largely to the fact that maximal chains in the nth Tamari lattice Tn range in length from n−1 to n2. In this note, we treat vertices in the lattice as Young diagrams and identify maximal chains as certain tableaux. For each i≥−1, we define Ci(n) as the set of maximal chains in Tn of length n+i. We give a recursion for #Ci(n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in Tn is #C0(n)=n3. The formula has a combinatorial interpretation in terms of a special property of maximal chains.