Lattice Paths, Young Tableaux, and Weight Multiplicities

For $\mathrm{\ell }\ge 1$ and $k\ge 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\mathrm{\ell }\times \mathrm{\ell }$ 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 $\mathrm{\ell }$ with height $\le k$, which is also the number of $(k+1)k\cdots 21$-avoiding permutations of $\{1,2,\dots ,\mathrm{\ell }\}$. 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 $V(k{\mathrm{\Lambda }}_0)$.