Generating Trees Unifying Several Classes of Four-Letter Pattern-Avoiding Descent Sequences

A descent sequence is a word π = π₁π₂ · · · πn of nonnegative integers satisfying π₁ = 0 and π_i ≤ 1 + des(π₁π₂ · · · π_i-1) for i = 2, 3, . . . , n, where des(π₁π₂ · · · π_m) denotes the number of indices j such that π_j > π_j+1. 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.