Lattice Paths, Young Tableaux, and Weight Multiplicities

For and , we consider certain admissible sequences of k-1 lattice paths in a colored 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 shape with , which is also the number of (k + 1)k center dot center dot center dot 21-avoiding permutations in . Finally, we apply this result to the representation theory of the affine Lie algebra and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight -module .