  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 $n$-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 as
Young 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 initial
values. The formula is a polynomial in $n$ of degree $3i+3$. For example, the
number of maximal chains of length $n$ in $\mathcal{T}_{n}$ is
$\#\mathcal{C}_{0}(n)={n \choose 3}$. The formula has a combinatorial
interpretation in terms of a special property of maximal chains.
