Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th