1,133 research outputs found
Permutations with restricted patterns and Dyck paths
We exhibit a bijection between 132-avoiding permutations and Dyck paths.
Using this bijection, it is shown that all the recently discovered results on
generating functions for 132-avoiding permutations with a given number of
occurrences of the pattern follow directly from old results on the
enumeration of Motzkin paths, among which is a continued fraction result due to
Flajolet. As a bonus, we use these observations to derive further results and a
precise asymptotic estimate for the number of 132-avoiding permutations of
with exactly occurrences of the pattern . Second,
we exhibit a bijection between 123-avoiding permutations and Dyck paths. When
combined with a result of Roblet and Viennot, this bijection allows us to
express the generating function for 123-avoiding permutations with a given
number of occurrences of the pattern in form of a continued
fraction and to derive further results for these permutations.Comment: 17 pages, AmS-Te
The area above the Dyck path of a permutation
In this paper we study a mapping from permutations to Dyck paths. A Dyck path
gives rise to a (Young) diagram and we give relationships between statistics on
permutations and statistics on their corresponding diagrams. The distribution
of the size of this diagram is discussed and a generalisation given of a parity
result due to Simion and Schmidt. We propose a filling of the diagram which
determines the permutation uniquely. Diagram containment on a restricted class
of permutations is shown to be related to the strong Bruhat poset.Comment: 9 page
Restricted Dumont permutations, Dyck paths, and noncrossing partitions
We complete the enumeration of Dumont permutations of the second kind
avoiding a pattern of length 4 which is itself a Dumont permutation of the
second kind. We also consider some combinatorial statistics on Dumont
permutations avoiding certain patterns of length 3 and 4 and give a natural
bijection between 3142-avoiding Dumont permutations of the second kind and
noncrossing partitions that uses cycle decomposition, as well as bijections
between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths.
Finally, we enumerate Dumont permutations of the first kind simultaneously
avoiding certain pairs of 4-letter patterns and another pattern of arbitrary
length.Comment: 20 pages, 5 figure
Permutations and Pairs of Dyck Paths
We define a map v between the symmetric group Sn and the set of pairs of Dyck paths of semilength n. We show that the map v is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map
Cycles and sorting index for matchings and restricted permutations
We prove that the Mahonian-Stirling pairs of permutation statistics (\sor,
\cyc) and (\inv, \mathrm{rlmin}) are equidistributed on the set of
permutations that correspond to arrangements of non-atacking rooks on a
Ferrers board with rows and columns. The proofs are combinatorial and
use bijections between matchings and Dyck paths and a new statistic, sorting
index for matchings, that we define. We also prove a refinement of this
equidistribution result which describes the minimal elements in the permutation
cycles and the right-to-left minimum letters. Moreover, we define a sorting
index for bicolored matchings and use it to show analogous equidistribution
results for restricted permutations of type and .Comment: 23 page
Avoiding patterns in irreducible permutations
International audienceWe explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index such that . The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length and the sets of irreducible permutations of length (respectively fixed point free irreducible involutions of length ) avoiding a pattern for . This induces two new bijections between the set of Dyck paths and some restricted sets of permutations
Statistics on pattern-avoiding permutations
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 111-116).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics 'number of fixed points' and 'number of excedances' is the same in 321-avoiding as in 132-avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statistics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of non-rational generating functions. Next we present a new statistic-preserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations.(cont.) Then we introduce a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. A part of our bijection is based on the Robinson-Schensted-Knuth correspondence. We also show that our bijection preserves additional parameters. Next, motivated by these results, we study the distribution of fixed points and excedances in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions which enumerate them. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique consists in using bijections between pattern-avoiding permutations and certain kinds of Dyck paths, in such a way that the statistics in permutations that we consider correspond to statistics on Dyck paths which are easier to enumerate. Finally, we study another kind of restricted permutations, counted by the Motzkin numbers. By constructing a bijection into Motzkin paths, we enumerate them with respect to some parameters, including the length of the longest increasing and decreasing subsequences and the number of ascents.by Sergi Elizalde.Ph.D
Pattern-avoiding even and odd Grassmannian permutations
In this paper, we investigate pattern avoidance of parity restricted (even or
odd) Grassmannian permutations for patterns of sizes 3 and 4. We use a
combination of direct counting and bijective techniques to provide recurrence
relations, closed formulas, and generating functions for their corresponding
enumerating sequences. In addition, we establish some connections to Dyck
paths, directed multigraphs, weak compositions, and certain integer partitions.Comment: 17 pages. Final version accepted for publicatio
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