0132-Avoiding Ascent Sequences

An ascent sequence of length n is a sequence $a_1{a}_2\cdots a_n$ of non-negative integers satisfying $a_1=0$ and, for $1<i\le n$, $a_i\le \mathrm{asc}(a_1{a}_2\cdots a_{i-1})+1$, where $\mathrm{asc}(a_1{a}_2\cdots a_k)$ denotes the number of ascents in the sequence $a_1{a}_2\cdots a_k$. 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.