Pattern avoidance in compositions and multiset permutations

Abstract

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 = 1 a 1 … k a k is a given multiset then the number of permutations of M that avoid the pattern ( 123 ) is a symmetric function of the multiplicities a 1 , … , a k . The bijection uses the Greene–Kleitman symmetric chain decomposition of the Boolean lattice.