Pattern avoidance in compositions and multiset permutations

One of the most arresting phenomena in the theory of pattern avoidance by permutations is the fact that the number of permutations of n letters that avoid a pattern π of 3 letters is independent of π. In this note we exhibit two generalizations of this fact, to ordered partitions, a.k.a. compositions, of an integer, and to permutations of multisets. It is remarkable that the conclusions are in those cases identical to those of the original case. Further, the number of permutations of a multiset S = 1122 . . . kk that avoid a given pattern π ∈ S3 is a symmetric function of the ai’s, and we will give here a bijective proof of this fact for π = (123). By a composition of an integer n into k parts we mean an integer representation