Enumeration of Inversion Sequences Avoiding 010 and Some Patterns of Length Four

We study classes of pattern-avoiding inversion sequences defined by simultaneously avoiding the classical pattern $010$ and one additional pattern $\tau$ of length four. We construct explicit generating trees that capture the structural evolution of these classes. For each of the studied classes, we derive functional equations for the corresponding generating functions, solve them explicitly in several cases, and compute closed-form or recursive formulas for the enumeration sequences. Using this framework, we obtain exact enumeration results for $I_n(010,\tau)$ when $\tau$ belongs to $\{0011, 0021, 0132, 0211, 0212, 0321, 0312\}$.