The Enumeration of Maximally Clustered Permutations

2005

In this work we characterize the intersections of these lattice paths and relate them to generalized barred permutation pattern avoidance.

Discrete Mathematics & Theoretical Computer Science, 2004

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

2019

The problem of avoiding a single pattern or a pair of patterns of length four by permutations has been well studied. Less is known about the avoidance of three 4-letter patterns. In this paper, we show that the number of members of Sn avoiding any one of twelve triples of 4-letter patterns is given by sequence A129775 in OEIS, which is known to count maximally clustered permutations. Numerical evidence confirms that there are no other (non-trivial) triples of 4letter patterns giving rise to this sequence and hence one obtains the largest (4, 4, 4)-Wilf-equivalence class for permutations. We make use of a variety of methods in proving our result, including recurrences, the kernel method, direct counting, and bijections.

Discrete Mathematics, 2005

2002

Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating functions for the number of permutations on n letters avoiding a generalized pattern ab-c where (a, b, c) ∈ S 3 , and containing a prescribed number of occurrences of generalized pattern cd-e where (c, d, e) ∈ S 3. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.

Arxiv preprint math/0107044, 2001

Abstract: Babson and Steingr\imsson 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 ...

Discrete Mathematics & Theoretical Computer Science, 2016

We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index i such that σ(i + 1) -σ(i) = 1. 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 n -1 and the sets of irreducible permutations of length n (respectively fixed point free irreducible involutions of length 2n) avoiding a pattern α for α ∈ {132, 213, 321}. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.

We investigate the structure of the two permutation classes defined by the sets of forbidden patterns {1234,2341} and {1243,2314}. By considering how the Hasse graphs of permutations in these classes can be built from a sequence of rooted source graphs, we determine their algebraic generating functions. Our approach is similar to that of "adding a slice", used previously to enumerate various classes of polyominoes and other combinatorial structures. To solve the relevant functional equations, we make extensive use of the kernel method.

To flatten a permutation expressed as a product of disjoint cycles, we mean to form another permutation by erasing the parentheses which enclose the cycles of the original. This clearly depends on how the cycles are listed. For permutations written in the standard cycle form-cycles arranged in increasing order of their first entries, with the smallest element first in each cycle-we count the permutations of [n] whose flattening avoids any subset of S3. Among the sequences that arise are central binomial coefficients, Schröder numbers, and relatives of the Fibonacci numbers. In some instances, we provide combinatorial arguments of the result, while in others, our approach is more algebraic. In a couple of the cases, we define an explicit bijection between the subset of Sn in question and a restricted set of lattice paths. In another, to establish the result, we make use of the kernel method to solve a functional equation arising once a certain parameter has been considered.

2013

Abstract. We characterize the sets of centrosymmetric permutations, namely, permutations σ ∈ Sn such that σ(i)+σ(n+1−i) = n+1, that avoid any given family of patterns of length 3. We exhibit bijections between some sets of restricted centrosymmetric permutations and sets of classical combinatorial objects, such as Dyck prefixes and subsets of [n] containing no consecutive integers.