Three families of pattern-avoiding ascent sequences and generating trees

An ascent sequence is a sequence of non-negative integers a₁a₂⋯an satisfying a₁ = 0 and, for 1 < i ≤ n, ai ≤ asc (a₁a₂⋯ai − 1) + 1,where asc (a₁a₂⋯ak) denotes the number of ascents in the sequence a₁a₂⋯ak. In this work, we provide alternative proofs for two known enumeration results: the number of ascent sequences of length n that avoid the pattern 0012 corresponds to the nth Catalan number, while those avoiding 0112 are counted by (3 n − 1 + 1)/2. Furthermore, we investigate the generating function associated with ascent sequences of length n avoiding the pattern 0122.