Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families $F_1 \subset F_2 \subset F_3 \subset F_4$ of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family $F_i$ which generalizes the one for the family $F_{i-1}$. The second topic of the paper is the enumeration of a fifth family $F_5$ of pattern-avoiding inversion sequences (containing $F_4$). This enumeration is also solved \emph{via} a succession rule, which however does not generalize the one for $F_4$. The associated enumeration sequence, which we call the \emph{powered Catalan numbers}, is quite intriguing, and further investigated. We provide two different succession rules for it, denoted $\Omega_{pCat}$ and $\Omega_{steady}$, and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the \emph{steady paths}, which are naturally associated with $\Omega_{steady}$. They allow us to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated with $\Omega_{pCat}$). Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures.