Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

We say that a permutation $\pi$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $\pi_a<\pi_b<\pi_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.Comment: 18 pages, 2 figure

On the diagram of 132-avoiding permutations

The diagram of a 132-avoiding permutation can easily be characterized: it is simply the diagram of a partition. Based on this fact, we present a new bijection between 132-avoiding and 321-avoiding permutations. We will show that this bijection translates the correspondences between these permutations and Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively, to each other. Moreover, the diagram approach yields simple proofs for some enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde

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

Variations on Hammersley's interacting particle process

The longest increasing subsequence problem for permutations has been studied extensively in the last fifty years. The interpretation of the longest increasing subsequence as the longest 21-avoiding subsequence in the context of permutation patterns leads to many interesting research directions. We introduce and study the statistical properties of Hammersleytype interacting particle processes related to these generalizations and explore the finer structures of their distributions. We also propose three different interacting particle systems in the plane analogous to the Hammersley process in one dimension and obtain estimates for the asymptotic orders of the mean and variance of the number of particles in the systems.Comment: 6 pages, 6 figures, accepted for publication in Discrete Mathematics Letter

Generating Permutations with Restricted Containers

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions

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

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