0132-Avoiding Ascent Sequences

An ascent sequence of length n is a sequence a1a2…an of non-negative integers 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 paper, we investigate ascent sequences avoiding the pattern 0132. By refining the structure of such sequences, we derive a generating tree and recurrence relations that characterize their enumeration. These recurrences are then translated into functional equations for the associated generating functions, which are solved using generating-function techniques. As a consequence, we obtain an explicit formula for the generating function that enumerates 0132-avoiding ascent sequences according to length.