Dense Packing of Patterns in a Permutation

Theoretical Computer Science, 2011

An infinite permutation α is a linear ordering of N. We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we define maximal pattern complexity p * α (n) for infinite permutations and show that this complexity function is ultimately constant if and only if the permutation is ultimately periodic; otherwise its maximal pattern complexity is at least n, and the value p * α (n) ≡ n is reached exactly on the family of permutations constructed by Sturmian words.

Electronic Journal of Combinatorics, 2021

We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern counts, both in terms of a permutation and in terms of its image under the fundamental bijection. We use these enumerations to resolve the question of characterizing socalled "shallow" permutations, whose depth (equivalently, disarray/displacement) is minimal with respect to length and reflection length. We present this characterization in several ways, including vincular patterns, mesh patterns, and a new object that we call "arrow patterns." Furthermore, we specialize to characterizing and enumerating shallow involutions and shallow cycles, encountering the Motzkin and large Schröder numbers, respectively.

arXiv (Cornell University), 2023

Let πn be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, the authors of showed that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2, . . . , n} in πn is n 2 2 (1o(1)) as n → ∞. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.

We study the number of permutations in the symmetric group on n elements that avoid con- secutive patterns S. We show that the spectrum of an associated integral operator on the space L2(0,1)m determines the asymptotic behavior of such permutations. Moreover, using an operator version of the clas- sical Frobenius-Perron theorem due to Kreùõn and Rutman, we prove asymptotic results for large classes of patterns S. This extends previously known results of Elizalde.

We provide upper and lower bounds for the expected length $\mathbb E(L_{n,m})$ of the longest common pattern contained in $m$ random permutations of length $n$. We also address the tightness of the concentration of $L_{n,m}$ around $\mathbb E(L_{n,m})$.

Journal of Difference Equations and Applications, 2013

We consider the problem of counting the occurrences of patterns of the form xy-z within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on Sn for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a unified approach to obtain explicit formulas. By these recurrences, we are able to determine simple closed form expressions for the number of permutations that, when flattened, avoid one of these patterns as well as expressions for the average number of occurrences. In particular, we find that the average number of 23-1 patterns and the average number of 32-1 patterns in Flatten(π), taken over all permutations π of the same length, are equal, as are the number of permutations avoiding either of these patterns. We also find that the average number of 21-3 patterns in Flatten(π) over all π is the same as it is for 31-2 patterns.

arXiv (Cornell University), 2023

In this paper, we study pattern avoidance for stabilized-interval-free (SIF) permutations. These permutations are contained in the set of indecomposable permutations and in the set of derangements. We enumerate pattern-avoiding SIF permutations for classical and pairs of patterns of size 3. In particular, for the patterns 123 and 231, we rely on combinatorial arguments and the fixed-point distribution of general permutations avoiding these patterns. We briefly discuss 123-avoiding permutations with two fixed points and offer a conjecture for their enumeration by the distance between their fixed points. For the pattern 231, we also give a direct argument that uses a bijection to ordered forests.

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.

Pure Mathematics and Applications

Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2,…, n} in π n is n 2 2 ( 1 - o ( 1 ) ) {{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right) as n → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.

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.