  For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of
$k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the
number of such admissible sequences of lattice paths is given by the sum of
squares of the number of standard Young tableaux of partitions of $\ell$ with
height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding
permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the
representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that
this quantity gives the multiplicity of certain maximal dominant weights in the
irreducible module $V(k\Lambda_0)$.
