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The Tamari lattices have been intensely studied since their introduction byDov Tamari around 1960. However oddly enough, a formula for the number ofmaximal chains is still unknown. This is due largely to the fact that maximalchains in the $n$-th Tamari lattice $\mathcal{T}_{n}$ range in length from$n-1$ to ${n \choose 2}$. In this note, we treat vertices in the lattice asYoung diagrams and identify maximal chains as certain tableaux. For each$i\geq-1$, we define $\mathcal{C}_{i}(n)$ as the set of maximal chains in$\mathcal{T}_{n}$ of length $n+i$. We give a recursion for$\#\mathcal{C}_{i}(n)$ and an explicit formula based on predetermined initialvalues. The formula is a polynomial in $n$ of degree $3i+3$. For example, thenumber of maximal chains of length $n$ in $\mathcal{T}_{n}$ is$\#\mathcal{C}_{0}(n)={n \choose 3}$. The formula has a combinatorialinterpretation in terms of a special property of maximal chains.

A recursion on maximal chains in the Tamari lattices