Pattern Avoidance Research Articles

A joint encryption and error correction scheme based on chaos and LDPC

Although some cryptosystems based on error correction code have been presented, the security and efficiency must be overcome before it can be realized. In this paper, a novel method of combined encryption and error correction based on hyperchaotic system and lower-density parity check (LDPC) code is proposed to provide both of the security and efficiency. The proposed system adopts a pseudorandom sequence generator based on hyperchaotic system for scrambling the plaintext and constructing the dynamic permutation box. The message is encoded by the LDPC encoder after it was scrambled and then encrypted by the permutation box. Different permutation patterns generated for different message blocks help to provide high security, while the encoder helps to provide capacity of error correction. MATLAB simulations reveal that the proposed scheme is more secure and effective than the existing joint encryption and error correction coding scheme. Moreover, the full error correction ability of LDPC is kept without confliction. So, the proposed scheme is suitable for secure communication system.

Governing singularities of symmetric orbit closures

We develop interval pattern avoidance and Mars-Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup GL_p x GL_q acting on the classical flag variety. We prove that all reasonable singularity properties can be classified in terms of interval patterns of clans.

Algorithmic and algebraic aspects of unshuffling permutations

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square (213,231)-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on directed perfect matchings, we prove that recognizing square permutations is NP-complete.

1324-avoiding permutations revisited

We give an improved algorithm for counting the number of 1324-avoiding permutations, resulting in 14 further terms of the generating function, which is now known for all lengths ≤50. We re-analyse the generating function and find additional evidence for our earlier conclusion that unlike other classical length-4 pattern-avoiding permutations, the generating function does not have a simple power-law singularity, but rather, the number of 1324-avoiding permutations of length n behaves asB⋅μn⋅μ1n⋅ng. We estimate μ=11.600±0.003, μ1=0.0400±0.0005, g=−1.1±0.1 while the estimate of B depends sensitively on the precise value of μ, μ1 and g. This reanalysis provides substantially more compelling arguments for the presence of the stretched exponential term μ1n.

Precedence permutation patterns creating criticality constellations: Exploring a conjecture on nonlinear activities with continuous links

Abstract The inaugural challenge of the 2016 Creative Construction Conference has posed two related questions on how many possible criticality constellations with different behaviors for delays and acceleration exist and how said constellations can occur for nonlinearly and monotonously progressing activities that have continuous relations. This paper systematically solves these questions by performing a thorough literature review, assembling theoretical foundations for link constellations, performing a computer simulation of all possible permutations, and providing a mathematical proof by contradiction. It is found that (for the initially assumed self-contained activities in a network schedule that exhibit only a linearly growing production), three newly hypothesized criticality constellations cannot exist. Nonlinear activity constellations with diverging or converging relative pro­ductivities are examined next. Lags in networks become buffers in linear schedules. It is found that a nonlinear curvature of the progress may induce middle-to-middle relations besides those between start and finish. If multiple curvatures are allowed, then partial segments can form relations, which increase the number of criticality constellations. This paper is extended from the 2017 Procedia Engineering conference version.

Semi-Baxter and Strong-Baxter: Two Relatives of the Baxter Sequence

In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern $2\underbracket{41}3$, which we call semi-Baxter permutations, and those avoiding the ...

Permutation groups, pattern involvement, and Galois connections

There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $S_\ell$ and for any $n \geq \ell$, the set of $n$-permutations involving only members of $G$ as $\ell$-patterns is a subgroup of $S_n$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels as automorphism groups of relations.

A sextuple equidistribution arising in Pattern Avoidance

We construct an intriguing bijection between 021-avoiding inversion sequences and (2413,4213)-avoiding permutations, which proves a sextuple equidistribution involving double Eulerian statistics. Two interesting applications of this result are also presented. Moreover, this result inspires us to characterize all permutation classes that avoid two patterns of length 4 whose descent polynomial equals that of separable permutations.

Pattern avoidance and fiber bundle structures on Schubert varieties

We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split pattern and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 3|12 and 23|1. Continuing, we show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241. This extends a combined result of Lakshmibai–Sandhya, Ryan, and Wolper who prove that Schubert varieties whose permutation avoids the “smooth” patterns 3412 and 4231 are iterated fiber bundles of smooth Grassmannian Schubert varieties.

Schur $P$-positivity and Involution Stanley Symmetric Functions

AbstractThe involution Stanley symmetric functions$\hat{F}_y$ are the stable limits of the analogs of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words and are indexed by the involutions in the symmetric group. By construction, each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila–Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials.

Longest Monotone Subsequences and Rare Regions of Pattern-Avoiding Permutations

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permutations. In addition, for a pattern $\tau$ of length $k$, we scale the plot of a random $\tau$-avoiding permutation down to the unit square and study the "rare region", which is the part of the square that is exponentially unlikely to contain any points. We prove that when $\tau_1>\tau_k$, the complement of the rare region is a closed set that contains the main diagonal of the unit square. For the case $\tau_1=k,$ we also show that the lower boundary of the part of the rare region above the main diagonal is a curve that is Lipschitz continuous and strictly increasing on $[0,1]$.

Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations

We prove that the multiplicities of certain maximal weights of $\mathfrak{g}(A^{(1)}_{n})$-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Misra-Rebecca. We also prove similar phenomena in types $A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. Both proofs are applications of Kashiwara's crystal theory.

Unimodal Permutations and Almost-Increasing Cycles

In this paper, we establish a natural bijection between the almost-increasing cyclic permutations of length $n$ and unimodal permutations of length $n-1$. This map is used to give a new characterization, in terms of pattern avoidance, of almost-increasing cycles. Additionally, we use this bijection to enumerate several statistics on almost-increasing cycles. Such statistics include descents, inversions, peaks and excedances, as well as the newly defined statistic called low non-inversions. Furthermore, we refine the enumeration of unimodal permutations by descents, inversions and inverse valleys. We conclude this paper with a theorem that characterizes the standard cycle notation of almost-increasing permutations.

Extremal functions of forbidden multidimensional matrices

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A d-dimensional zero–one matrix A avoids anotherd-dimensional zero–one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to study the maximum number of nonzero entries in a d-dimensional n×⋯×n matrix that avoids P. This maximum number, denoted by f(n,P,d), is called the extremal function.We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis. Firstly, we obtain non-trivial lower and upper bounds on f(n,P,d) when n is large for every d-dimensional block permutation matrix P. We establish the tight bound Θ(nd−1) on f(n,P,d) for every d-dimensional tuple permutation matrix P. This tight bound has the lowest possible order that an extremal function of a nontrivial matrix can ever achieve. Secondly, we show that the limit inferior of the sequence {f(n,P,d)nd−1} has a lower bound 2Ω(k1∕2) for a family of k×⋯×k permutation matrices P. We also improve the upper bound on the limit superior from 2O(klogk) to 2O(k) for all k×⋯×k permutation matrices and show that the new upper bound also holds for tuple permutation matrices.

Algorithms for testing occurrences of length 4 patterns in permutations

In this paper linear time sequential and optimal parallel algorithms for testing pattern involvement for all length 4 permutations are described. This is an improvement as the previous best sequential algorithms, for most of these pattern require $$O(n\log n)$$ time. Our parallel algorithms can be implemented in $$O(\log n)$$ time with $$n/\log n$$ processors on the CREW PRAM model, or alternatively in $$O(\log \log \log n)$$ time with $$n/\log \log \log n$$ processors on a CRCW PRAM PRAM model. Parallel algorithm can also be implemented in constant time with $$n\log ^3 n$$ processors on a CRCW PRAM model. The previous best parallel algorithms were available for only some of these patterns and took $$O(\log n)$$ time with n processors on the CREW PRAM model.

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.

Fast algorithms for finding pattern avoiders and counting pattern occurrences in permutations

Given a set $\Pi$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(\Pi)$, the set of permutations of length at most $n$ avoiding the patterns in $\Pi$, in time $O(|S_{\le n - 1}(\Pi)| \cdot k + |S_{n}(\Pi)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $\Pi$ in each permutation in $S_n$. Surprisingly, when $|\Pi| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutation analyzed, all of our algorithms take time at most polynomial per outputted permutation. If we want to solve only the enumerative variant of each problem, computing $|S_{\le n}(\Pi)|$ or tallying permutations according to $\Pi$-patterns, rather than to store information about every permutation, then all of our algorithms can be implemented in $O(n^{k+1}k)$ space. Using our algorithms, we generated $|S_5(\Pi)|, \ldots, |S_{16}(\Pi)|$ for each $\Pi \subseteq S_4$ with $|\Pi| > 4$, and analyzed OEIS matches. We obtained a number of potentially novel pattern-avoidance conjectures. Our algorithms extend to considering permutations in any set closed under standardization of subsequences. Our algorithms also partially adapt to considering vincular patterns.

Avoidance of Type (1,2) Patterns by Catalan Words

A certain subset of the multiset permutations of length n satisfying two restrictions has been recently shown to be enumerated by the Catalan number Cn−1. These sequences have been termed Catalan words and are closely related to the 321-avoiding permutations. Here, we consider the problem of avoidance of patterns of type (1,2) wherein the second and third letters within an occurrence of a pattern are required to be adjacent. We derive in several cases functional equations satisfied by the generating functions enumerating members of the avoidance class which we solve by various methods. In one case, the generating function can be expressed in terms of a sum of reciprocals of Chebyshev polynomials, while in another, in terms of a previously studied q-Bell number. Among the sequences arising as enumerators of avoidance classes are the Motzkin and Fibonacci numbers. In several cases, it is more convenient to consider first the problem of avoidance on the subset of Catalan words whose members have no adjacent letters the same before moving to the larger problem on all Catalan words.

Nine classes of permutations enumerated by binomial transform of Fine’s sequence

The problem of avoidance of a single permutation pattern or of a pair of patterns of length four has been well studied. Less is known concerning the avoidance of three 4-letter patterns. In this paper, we determine up to symmetry all triples of 4-letter patterns such that the number of members of Sn avoiding any one of them is given by the binomial transform of Fine’s sequence (see A033321 in OEIS). We make use of both algebraic and combinatorial proofs in order to establish our results. In a couple of cases, we introduce certain auxiliary statistics on Sn which give rise to a system of functional equations that can be solved using the kernel method. In another case, a direct bijection is defined between members of the avoidance class in question and the set of skew Dyck paths.

A New Bijective Proof of Babson and Steingrímsson's Conjecture

Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingrímsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.