We determine explicitly the maximal dominant weights for the integrable highest weight sl(n)-modulesWe give a conjecture for the number of maximal dominant weights of V (kΛ 0 ) and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of V (kΛ 0 ). We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for k = 2 and give computational evidence for the validity of this conjecture for k > 2.

For ℓ ≥ 1 and k ≥ 2, we consider certain admissible sequences of k−1 lattice paths in a colored ℓ×ℓ 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 ℓ with height ≤ k, which is also the number of (k + 1)k · · · 21-avoiding permutations of {1, 2, . . . , ℓ}. Finally, we apply this result to the representation theory of the affine Lie algebra sl(n) and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module V (kΛ 0 ).2010 Mathematics Subject Classification. Primary 05E10, 17B10; Secondary 05A05, 05A17, 17B67.Key words and phrases. Lattice path and Young tableau and avoiding permutation and affine Lie algebra and weight multiplicity.

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Consider the affine Lie algebra sℓ(n) with null root δ, weight lattice P and set of dominant weightsmax + (kΛ 0 ) = max(kΛ 0 ) ∩ P + denote the set of maximal dominant weights which is known to be a finite set. In 2014, the authors gave the complete description of the set max + (kΛ 0 ) [2] . In subsequent papers [3,4], the multiplicities of certain subsets of max + (kΛ 0 ) were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the sℓ(n)-module V (kΛ 0 ) are given generalizing the results in [3,4].