Pattern avoidance in compositions and multiset permutations

We show that among the compositions of n into positive parts, the number g(n) that avoid a given pattern π of three letters is independent of π. We find the generating function of {g(n)}, and it shows that the sequence {g(n)} is not P-recursive. If S is a given multiset, we show that the number of permutations of S that avoid a pattern π of three letters is independent of π. Finally, we give a bijective proof of the fact that if M=a11…kak is a given multiset then the number of permutations of M that avoid the pattern (123) is a symmetric function of the multiplicities a1,…,ak. The bijection uses the Greene–Kleitman symmetric chain decomposition of the Boolean lattice.