A descent sequence is a word of nonnegative integers satisfying that the first term is zero, and each subsequent term is at most one plus the number of descents among the preceding terms, where a descent is a position at which a term is greater than the term immediately following it. In this work, we study descent sequences subject to the additional restriction of avoiding a given pattern of length four. We analyze seven distinct avoidance classes and provide enumerative results for each of them. Our approach is based on the construction of generating trees with one or two labels, from which we derive succession rules and corresponding systems of recurrence relations. These recurrences are then used to compute explicit generating functions for the number of descent sequences of length n avoiding either 0001, 0010, 0011, 0012, 0021, 0110, 0112, 0123, or 0132.
Toufik Mansour (Tue,) studied this question.