Subset Of Permutations Research Articles

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.

Permutation Tests Using Arbitrary Permutation Distributions

Permutation tests date back nearly a century to Fisher’s randomized experiments, and remain an immensely popular statistical tool, used for testing hypotheses of independence between variables and other common inferential questions. Much of the existing literature has emphasized that, for the permutation p-value to be valid, one must first pick a subgroup G of permutations (which could equal the full group) and then recalculate the test statistic on permuted data using either an exhaustive enumeration of G, or a sample from G drawn uniformly at random. In this work, we demonstrate that the focus on subgroups and uniform sampling are both unnecessary for validity—in fact, a simple random modification of the permutation p-value remains valid even when using an arbitrary distribution (not necessarily uniform) over any subset of permutations (not necessarily a subgroup). We provide a unified theoretical treatment of such generalized permutation tests, recovering all known results from the literature as special cases. Thus, this work expands the flexibility of the permutation test toolkit available to the practitioner.

Algorithmic coincidence classification of mesh patterns

Permutations have connections to other mathematical objects such as Schubert varieties, sorting networks, and genome rearrangements. Often the connection is described in terms of patterns that are absent from the permutations. There can be ambiguity in this description, in the sense that the same subset of permutations can be defined with two different patterns. This is called coincidence and we focus on the coincidence of mesh patterns, one of the most descriptive version of patterns in permutations. We review and extend previous results on coincidence of mesh patterns. We introduce the notion of a force on a permutation pattern and apply it to the coincidence classification of mesh patterns, completing the classification up to size three. We also show that this concept can be used to enumerate classical permutation classes.

Permutation-Based Block Code for Short Packet Communication Systems.

This paper presents an approach to the construction of block error-correcting code for data transmission systems with short packets. The need for this is driven by the necessity of information interaction between objects of machine-type communication network with a dynamically changing structure and unique system of commands or alerts for each network object. The codewords of a code are permutations with a given minimum pairwise Hamming distance. The purpose of the study is to develop a statistical method for constructing a code, in contrast to known algebraic methods, and to investigate the code size. An algorithm for generating codewords has been developed. It can be implemented both by enumeration of the full set of permutations, and by enumeration of a given number of randomly selected permutations. We have experimentally determined the dependencies of the average and the maximum values of the code size on the size of a subset of permutations used for constructing the code. A technique for computing approximation quadratic polynomials for the determined code size dependencies has been developed. These polynomials and their corresponding curves estimate the size of a code generated from a subset of random permutations of such a size that a statistically significant experiment cannot be performed. The results of implementing the developed technique for constructing a code based on permutations of lengths 7 and 11 have been presented. The prediction relative error of the code size did not exceed the value of 0.72% for permutation length 11, code distance 9, random permutation subset size 50,000, and permutation statistical study range limited by 5040.

ОЦІНКА ВЛАСТИВОСТЕЙ СИНТЕЗОВАНИХ НА ОСНОВІ ТЕОРІЇ РЕШІТОК СИГНАЛЬНО-КОДОВИХ КОНСТРУКЦІЙ ДЛЯ НЕРОЗДІЛЬНИХ ФАКТОРІАЛЬНИХ КОДІВ

One of the main functions of computer systems and networks is to ensure the security of information circulating in the system. At the same time, an equally important function is to protect information from the effects of interference in the communication channel. The use of factorial codes is one approach to combining noise-tolerant encryption and cryptographic protection. But these codes are vulnerable to even errors, which lead to the transformation of one permutation from the allowed set to another permutation belonging to the same set. Therefore, the issue of improving the reliability of data transmission in systems with factorial coding is relevant. The purpose of the study is to evaluate the method of forming a signal-code structure, which ensures the achievement of the required value of transmission reliability and allows to maximize the code speed by forming a subset of permutations based on lattice theory. The relative transmission rate and the probability of undetected error as a result of the use of synthesized signal-code constructs in systems with integral factorial coding are subject to evaluation. To evaluate the efficiency of the synthesized signal-code constructions, the methods of factorial coding with data recovery by permutation and factorial coding with data recovery by permutation with error correction have been used. In order to determine the probability of undetected error, a software model has been developed that simulates a data transmission environment with independent bit errors. The results of the software model have made it possible to establish the dependence of the probability of undetected error in systems with integral factorial coding on the probability of bit error in channels with independent bit errors. A comparative analysis of the relative transmission rate and the probability of undetected error in systems with factorial coding with data recovery by permutation and factorial coding with data recovery by permutation with error correction is also performed. The results obtained in this paper have made it possible to determine the main advantages and disadvantages of the method of forming signal-code structures based on the lattice theory, as well as to determine the scope of this method.

Aldous’s spectral gap conjecture for normal sets

Let S n S_{n} denote the symmetric group on n n elements, and let Σ ⊆ S n \Sigma \subseteq S_{n} be a symmetric subset of permutations. Aldous’s spectral gap conjecture, proved by Caputo, Liggett, and Richthammer [J. Amer. Math. Soc. 23 (2010), no. 3, 831–851], states that if Σ \Sigma is a set of transpositions, then the second eigenvalue of the Cayley graph C a y ( S n , Σ ) \mathrm {Cay\!}\left (S_{n},\Sigma \right ) is identical to the second eigenvalue of the Schreier graph on n n vertices depicting the action of S n S_{n} on { 1 , … , n } \left \{ 1,\ldots ,n\right \} . Inspired by this seminal result, we study similar questions for other types of sets in S n S_{n} . Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [Invent. Math. 174 (2008), no. 3, 645–687], we show that for large enough n n , if Σ ⊂ S n \Sigma \subset S_{n} is a full conjugacy class, then the second eigenvalue of C a y ( S n , Σ ) \mathrm {Cay}\!\left (S_{n},\Sigma \right ) is roughly identical to the second eigenvalue of the Schreier graph depicting the action of S n S_{n} on ordered 4 4 -tuples of elements from { 1 , … , n } \left \{ 1,\ldots ,n\right \} . We further show that this type of result does not hold when Σ \Sigma is an arbitrary normal set, but a slightly weaker one does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set Σ ⊂ S n \Sigma \subset S_{n} , which yields surprisingly strong consequences.

Restricted permutations refined by number of crossings and nestings

Let st={st1,…,stk} be a set of k statistics on permutations with k≥1. We say that two given subsets of permutations T and T′ are st-Wilf-equivalent if the joint distributions of all statistics in st over the sets of T-avoiding permutations Sn(T) and T′-avoiding permutations Sn(T′) are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in S3, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection Θ:Sn(321)→Sn(132) which was originally exhibited by Elizalde and Pak in Elizalde and Pak (2004). They proved that the bijection Θ preserves the number of fixed points and excedances. Since the given formulation of Θ is not direct, we show that it can be defined directly by a recursive formula. Then, we prove that it also preserves the number of crossings. Due to the fact that the sets of non-nesting permutations and 321-avoiding permutations are the same, these properties of the bijection Θ lead to an unexpected result related to the q,p-Catalan numbers of Randrianarivony defined in Randrianarivony (1998).

Signed mahonian identities on permutations with subsequence restrictions

In this paper, we present a number of results surrounding Caselli's conjecture on the equidistribution of the major index with sign over the two subsets of permutations of {1,2,…,n} containing respectively the word 12⋯k and the word (n−k+1)⋯n as a subsequence, under a parity condition of n and k. We derive broader bijective results on permutations containing varied subsequences. As a consequence, we obtain the signed mahonian identities on families of restricted permutations, in the spirit of a well-known formula of Gessel–Simion, covering a combinatorial proof of Caselli's conjecture. We also derive an extension of the insertion lemma of Han and Haglund–Loehr–Remmel which allows us to obtain a signed enumerator of the major-index increments resulting from the insertion of a pair of consecutive numbers in any place of a given permutation.

On the combinatorics of commutators of Lie algebras

Motivated by the combinatorial properties of products in Lie algebras, we investigate the subset of permutations that naturally appears when we write the long commutator [Formula: see text] as a sum of associative monomials. We characterize this subset and find some useful equivalences. Moreover, we explore properties concerning the action of this subset on sequences of [Formula: see text] elements. In particular, we describe sequences that share some special symmetries which can be useful in the study of combinatorial properties in graded Lie algebras.

(q,t)-Catalan numbers: gamma expansions, pattern avoidances, and the (−1)-phenomenon

The aim of this paper is two-fold. We first prove several new interpretations of a kind of (q,t)-Catalan numbers along with their corresponding γ-expansions using pattern avoiding permutations. Secondly, we give a complete characterization of certain (−1)-phenomenon for each subset of permutations avoiding a single pattern of length three, and discuss their q-analogues utilizing the newly obtained q-γ-expansions, as well as the continued fraction of a quint-variate generating function due to Shin and the fourth author. Moreover, we enumerate the alternating permutations avoiding simultaneously two patterns, namely (2413,3142) and (1342,2431), of length four, and consider such (−1)-phenomenon for these two subsets as well.

Colouring of generalized signed triangle-free planar graphs

We view an undirected graph G as a symmetric digraph, where each edge xy is replaced by two opposite arcs e=(x,y) and e−1=(y,x). Assume S is an inverse closed subset of permutations of positive integers. We say G is S-k-colourable if for any mapping σ:E(G)→S with σ(x,y)=(σ(y,x))−1, there is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). The concept of S-k-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets S such that every triangle-free planar graph is S-3-colourable. Such a set S is called TFP-good. Grötzsch’s theorem is equivalent to say that S={id} is TFP-good. We prove that for any inverse closed subset S of S3 which is not isomorphic to {id,(12)}, S is TFP-good if and only if either S={id} or there exists a∈[3] such that for each π∈S, π(a)≠a. It remains an open question to determine whether or not S={id,(12)} is TFP-good.

On rotated Schur-positive sets

The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al. and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of the proof we obtain a notion of cyclic descent set on these tableaux, which is rotated by the cyclic action on them.

Snake-in-the-Box Codes for Rank Modulation under Kendall’s <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math> </inline-formula>-Metric in <inline-formula> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math> </inline-formula>

Snake-in-the-box codes under Kendall’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -metric are studied in the rank modulation scheme for flash memories, where codewords are a subset of permutations in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{n}$ </tex-math></inline-formula> with minimal Kendall’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -distance two, and two cyclically consecutive codewords are connected via a push-to-the-top operation. Studies so far restrict the push-to-the-top operations only on odd indices, resulting in a snake consisting of permutations with the same parity, and thus, the minimal distance constraint is easily satisfied. Asymptotically optimal snake codes have been constructed this way in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> . As for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> , this framework keeps the last element fixed, and thus, a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> is equivalent to a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> , which is rather trivial. If one wants to do better, then it is inevitable to have some push-to-the-top operations on even indices, resulting in a combination of odd and even permutations in the snake, which increases the difficulty to guarantee the minimal Kendall’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> -distance constraint. Thus, Horovitz and Etzion pose the open problem to prove or disprove that the size of the largest snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> is not larger than the size of the largest snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> . A first step toward this problem is a negative answer by Wang and Fu, who construct a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> with exactly one more permutation than an optimal snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+1}$ </tex-math></inline-formula> . In this paper, we give an explicit construction of a snake in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S_{2n+2}$ </tex-math></inline-formula> with size asymptotically approaching <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$({1}/{4})|S_{2n+2}|$ </tex-math></inline-formula> .

Buses, Bullies, and Bijections

SummaryThe random—or orderly—seating of passengers on a bus is used to motivate several questions about cycles of permutations. These in turn motivates the investigation of bijections between special subsets of permutations. The goal, of course, is to give simple explanations of surprising facts.

Reordering space design in statistical machine translation

In Statistical Machine Translation (SMT), the constraints on word reorderings have a great impact on the set of potential translations that is explored during search. Notwithstanding computational issues, the reordering space of a SMT system needs to be designed with great care: if a larger search space is likely to yield better translations, it may also lead to more decoding errors, because of the added ambiguity and the interaction with the pruning strategy. In this paper, we study the reordering search space, using a state-of-the art translation system, where all reorderings are represented in a permutation lattice prior to decoding. This allows us to directly explore and compare different reordering schemes and oracle settings. We also study in detail a rule-based preordering system, varying the length and number of rules, the tagset used, as well as contrasting with purely combinatorial subsets of permutations. We carry out experiments on three language pairs in both directions: English-French, a close language pair; English-German and English-Czech, two much more challenging pairs. We show that even though it might be desirable to design better reordering spaces, model and search errors seem to be the most important issues. Therefore, improvements of the reordering space should come along with improvements of the associated models to be really effective.

Pattern avoiding partitions and Motzkin left factors

Abstract Let L n, n ≥ 1, denote the sequence which counts the number of paths from the origin to the line x = n − 1 using (1, 1), (1, −1), and (1, 0) steps that never dip below the x-axis (called Motzkin left factors). The numbers L n count, among other things, certain restricted subsets of permutations and Catalan paths. In this paper, we provide new combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify four classes of the partitions of size n, all of which have cardinality L n and each avoiding a set of two classical patterns of length four. We obtain a further generalization in one of the cases by considering a pair of statistics on the partition class. In a couple of cases, to show the result, we make use of the kernel method to solve a functional equation arising after a certain parameter has been introduced.

Disjointness of interval exchange transformations from systems of probabilistic origin

We prove the disjointness ofalmost all interval exchange transformations from ELF systems(systems of probabilistic origin) for a countable subset ofpermutations including the symmetric permutations $ 1\ 2\ \ldots \ m-1 \ m $ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ m-1 \ldots \ 2\ 1 $ for m=3,5,7.Some disjointness properties of special flows built over intervalexchange transformations and under piecewise constant rooffunction are investigated as well.

Boolean functions of n variables and permutations on

For each Boolean function in n variables, from the expression of the product of all its Walsh spectrum values derived in a precedent paper, we deduce a new characterization of the parity of its distance from the set of all the affine functions. This characterization uses a subset of permutations on , and some new properties on this subset are deduced.

Impact of explosive eruption scenarios at Vesuvius

Impact of explosive eruption scenarios at Vesuvius

Counting Descents and Ascents Relative to Equivalence Classes mod k

Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in position i, i + 1, ... , i + j − 1 in σ have the same relative order as the elements of τ. If \(\Upsilon\) is the set of permutations of length j, then we say that a permutation σ has a τ-match starting at position j if it has a τ-match at position j for some \(\tau \in \Upsilon\). A number of recent papers have studied the distribution of τ-matches and \(\Upsilon\)-matches in permutations. In this paper, we consider a more refined pattern matching condition where we take into account conditions involving the equivalence classes of the elements mod k for some integer k ≥ 2. In this paper, we prove explicit formulas for the number of permutations of n which have s τ-equivalence mod k matches when τ is of length 2. We also show that similar formulas hold for \(\Upsilon\) -equivalence mod k matches for certain subsets of permutations of length 2.