5,555 research outputs found
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
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 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 which
generalizes the one for the family .
The second topic of the paper is the enumeration of a fifth family of
pattern-avoiding inversion sequences (containing ). This enumeration is
also solved \emph{via} a succession rule, which however does not generalize the
one for . 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 and
, 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 . 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
).
Along the way, we provide several nice connections to families of
permutations defined by the avoidance of vincular patterns, and some
enumerative conjectures.Comment: V2 includes modifications suggested by referees (in particular, a
much shorter Section 3, to account for arXiv:1706.07213
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