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Discrete Mathematics & Theoretical Computer Science |
vol. 26:1, Permutation Patterns 2023
This is a special issue following the 2023 edition of the international conference on Permutation Patterns conference, which was held in Dijon, France, July 3-7, 2023.
1. A logical limit law for $231$-avoiding permutations
We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis.
Section:
Special issues
2. Pattern Avoidance in Weak Ascent Sequences
In this paper, we study pattern avoidance in weak ascent sequences, giving some results for patterns of length 3. This is an analogous study to one given by Duncan and Steingr\'imsson (2011) for ascent sequences. More precisely, we provide systematically the generating functions for the number of weak ascent sequences avoiding the patterns $001, 011, 012, 021$, and $102$. Additionally, we establish bijective connections between pattern-avoiding weak ascent sequences and other combinatorial objects, such as compositions, upper triangular 01-matrices, and plane trees.
Section:
Special issues
3. Long increasing subsequences and non-algebraicity
We use a recent result of Alin Bostan to prove that the generating functions of two infinite sequences of permutation classes are not algebraic.
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Special issues
4. Joint distributions of statistics over permutations avoiding two patterns of length 3
Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics (asc, des, lrmax, lrmin, rlmax, rlmin), and also explicit formulas for the joint distribution of four statistics (asc, des, MNA, MND) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang of the statistics MNA and MND (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions.
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Special issues
5. An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation
Let $\pi_n$ 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 a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. 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.
Section:
Special issues
6. A positional statistic for 1324-avoiding permutations
We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for which $a\prec n\prec [a-1]$, $a\ge 1$. This notation means that, when written in one line notation, such a permutation must have $a$ to the left of $n$, and the elements of $\{1,\dots,a-1\}$ must all be to the right of $n$. For $n\ge 2$, we establish a connection between the subset of permutations in $S_n^{1\prec n}(1324)$ having the 1 adjacent to the $n$ (called primitives), and the set of 1324-avoiding dominoes with $n-2$ points. For $a\in\{1,2\}$, we introduce constructive algorithms and give formulas for the enumeration of $S_n^{a\prec n}(1324)$ by the position of $a$ relative to the position of $n$. For $a\ge 3$, we formulate some conjectures for the corresponding generating functions.
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Special issues
7. Permutation Entropy for Signal Analysis
Shannon Entropy is the preeminent tool for measuring the level of uncertainty (and conversely, information content) in a random variable. In the field of communications, entropy can be used to express the information content of given signals (represented as time series) by considering random variables which sample from specified subsequences. In this paper, we will discuss how an entropy variant, the \textit{permutation entropy} can be used to study and classify radio frequency signals in a noisy environment. The permutation entropy is the entropy of the random variable which samples occurrences of permutation patterns from time series given a fixed window length, making it a function of the distribution of permutation patterns. Since the permutation entropy is a function of the relative order of data, it is (global) amplitude agnostic and thus allows for comparison between signals at different scales. This article is intended to describe a permutation patterns approach to a data driven problem in radio frequency communications research, and includes a primer on all non-permutation pattern specific background. An empirical analysis of the methods herein on radio frequency data is included. No prior knowledge of signals analysis is assumed, and permutation pattern specific notation will be included. This article serves as a self-contained introduction to the relationship between permutation patterns, entropy, and signals analysis for studying radio frequency signals and […]
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Special issues
8. Interval and $\ell$-interval Rational Parking Functions
Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.
Section:
Combinatorics