8 research outputs found
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Inversion Polynomials for Permutations Avoiding Consecutive Patterns
In 2012, Sagan and Savage introduced the notion of -Wilf equivalence for
a statistic and for sets of permutations that avoid particular permutation
patterns which can be extended to generalized permutation patterns. In this
paper we consider -Wilf equivalence on sets of two or more consecutive
permutation patterns. We say that two sets of generalized permutation patterns
and are -Wilf equivalent if the generating function for the
inversion statistic on the permutations that simultaneously avoid all elements
of is equal to the generating function for the inversion statistic on the
permutations that simultaneously avoid all elements of .
In 2013, Cameron and Killpatrick gave the inversion generating function for
Fibonacci tableaux which are in one-to-one correspondence with the set of
permutations that simultaneously avoid the consecutive patterns and
In this paper, we use the language of Fibonacci tableaux to study the
inversion generating functions for permutations that avoid where is
a set of five or fewer consecutive permutation patterns. In addition, we
introduce the more general notion of a strip tableaux which are a useful
combinatorial object for studying consecutive pattern avoidance. We go on to
give the inversion generating functions for all but one of the cases where
is a subset of three consecutive permutation patterns and we give several
results for a subset of two consecutive permutation patterns
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
Transport of patterns by Burge transpose
We take the first steps in developing a theory of transport of patterns from
Fishburn permutations to (modified) ascent sequences. Given a set of pattern
avoiding Fishburn permutations, we provide an explicit construction for the
basis of the corresponding set of modified ascent sequences. Our approach is in
fact more general and can transport patterns between permutations and
equivalence classes of so called Cayley permutations. This transport of
patterns relies on a simple operation we call the Burge transpose. It operates
on certain biwords called Burge words. Moreover, using mesh patterns on Cayley
permutations, we present an alternative view of the transport of patterns as a
Wilf-equivalence between subsets of Cayley permutations. We also highlight a
connection with primitive ascent sequences.Comment: 24 pages, 4 figure
Polytopes, generating functions, and new statistics related to descents and inversions in permutations
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2008.Includes bibliographical references (p. 75-76).We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation [sigma] = [sigma] 1 [sigma] 2 an defined as the set of indices i such that either i is odd and ai > ui+l, or i is even and au < au+l. We show that this statistic is equidistributed with the 3-descent set statistic on permutations [sigma] = [sigma] 1 [sigma] 2 ... [sigma] n+1 with al = 1, defined to be the set of indices i such that the triple [sigma] i [sigma] i + [sigma] i +2 forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials ... using alternating descents. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new qanalog of the Euler number En and show how it emerges in a q-analog of an identity expressing E, as a weighted sum of Dyck paths. Other parts of this thesis are devoted to polytopes relevant to the descent statistic. One such polytope is a "signed" version of the Pitman-Stanley parking function polytope, which can be viewed as a generalization of the chain polytope of the zigzag poset. We also discuss the family of descent polytopes, also known as order polytopes of ribbon posets, giving ways to compute their f-vectors and looking further into their combinatorial structure.by Denis Chebikin.Ph.D
Enumerating permutations avoiding a pair of Babson-Steingrimsson patterns
Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1, 2) or (2, 1). We also conjecture the number of permutations avoiding the patterns in any set of three or more such patterns
Enumerating permutations avoiding a pair of Babson-Steingrimsson patterns
Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type (1, 2) or (2, 1). We also conjecture the number of permutations avoiding the patterns in any set of three or more such patterns