Generalized permutation patterns -- a short survey
Einar Steingrimsson
备注：
11 pages. Added a section on asymptotics (Section 8), added more examples of barred patterns equal to generalized patterns (Section 7) and made a few other minor additions. To appear in ``Permutation Patterns, St Andrews 2007'', S.A. Linton, N. Ruskuc, V. Vatter (eds.), LMS Lecture Note Series, Cambridge University Press
An occurrence of a classical pattern p in a permutation πis a subsequence of πwhose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance--or the prescribed number of occurrences--of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.
This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the cd-index of the symmetric group.
We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based.
A consecutive pattern in a permutation $π$ is another permutation $σ$ determined by the relative order of a subsequence of contiguous entries of $π$. Traditional notions such as descents, runs and peaks can be viewed as particular examples of consecutive patterns in permutations, but the systematic study of these patterns has flourished in the last 15 years, during which a variety of different techniques have been used. We survey some interesting developments in the subject, focusing on exact and asymptotic enumeration results, the classification of consecutive patterns into equivalence classe…
This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments.
A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if any two sets in $\mathcal{A}$ have at least $t$ common elements. A central problem in extremal set theory is to determine the size or structure of a largest $t$-intersecting sub-family of a given family $\mathcal{F}$. We give a survey of known results, conjectures and open problems for various important families $\mathcal{F}$, namely, power sets, levels of power sets, hereditary families, families of signed sets, families of labeled sets, and families of permutations. We also provide some extensions and consequences of kno…
The density, denoted by $κ(n,r)$, of permutations having no cycles of length less than $r+1$ in a symmetric group $\mathrm{S}_n$ is explored. New asymptotic formulas for $κ(n,r)$ are obtained using the saddle-point method when $5\leq r< n$ and $n\to\infty$.
We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most $\flr{2n/3}$ such moves and that some permutations require at least $\flr{n/2}$ moves.
The concept of pattern avoidance respectively containment in permutations can be extended to permutations on multisets in a straightforward way. In this note we present a direct proof of the already known fact that the well-known Stanley-Wilf Conjecture, stating that the number of permutations avoiding a given pattern does not grow faster than exponentially, also holds for permutations on multisets.
We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can be permuted to avoid 3-term and 4-term APs. We also show that any permutation of the positive integers must contain a 3-term AP with odd common difference as a subsequence, and construct a permutation of the positive integers that does not contain any 4-term AP with odd common difference.
This is a brief survey of some open problems on permutation patterns, with an emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns in Permutations and words}. I first survey recent developments on the enumeration and asymptotics of the pattern 1324, the last pattern of length 4 whose asymptotic growth is unknown, and related issues such as upper bounds for the number of avoiders of any pattern of length $k$ for any given $k$. Other subjects treated are the Möbius function, topological properties and other algebraic aspects of the poset of permutations, ordered by conta…
The $n$-th rencontres number with the parameter $r$ is the number of permutations having exactly $r$ fixed points. In particular, a derangement is a permutation without any fixed point. We presents a short combinatorial proof for a weighted sum derangement identities.
We present a short, direct proof of the fact that the generating function of all permutations of a fixed length $n\geq 4$ is divisible by $(1+z)^m$, where $m=\lfloor (n-2)/2 \rfloor$.
Distributions of Mesh Patterns of Short Lengths on Separable Permutations
Northwestern Polytechnical University（西北工业大学）；Shenzhen Research Institute of Northwestern Polytechnical University（西北工业大学深圳研究院）；University of Strathclyde（思克莱德大学）
Call a permutation $k$-inflatable if the sequence of its tensor products with uniform random permutations of increasing lengths has uniform $k$-point pattern densities. Previous work has shown that nontrivial $k$-inflatable permutations do not exist for $k \geq 4$. In this paper, we derive a general formula for the limit densities of patterns in the sequence of tensor products of a fixed permutation with each permutation from a convergent sequence. By applying this result, we completely characterize $3$-inflatable permutations and find explicit examples of $3$-inflatable permutations with vari…
Following Babai's algorithm for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups; this is feasible both in the original version of the algorithm and in its CFSG-free version, partially done by Babai and completed by Pyber. The existence of such descriptions gives a weak form of the Cameron-Maróti classification even without assuming CFSG. We also thoroughly explicate Babai's recursion process (as given in Helfgott) and obtain explicit constants for the runtime of the …
In this short note we present a simple combinatorial trick which can be effectively applied to show the non--existence of sharply transitive sets of permutations in certain finite permutation groups.
We present various results on multiplying cycles in the symmetric group. Our first result is a generalisation of the following theorem of Boccara (1980): the number of ways of writing an odd permutation in the symmetric group on $n$ symbols as a product of an $n$-cycle and an $n-1$-cycle is independent of the permutation chosen. We give a number of different approaches of our generalisation. One partial proof uses an inductive method which we also apply to other problems. In particular, we give a formula for the distribution of the number of cycles over all products of cycles of fixed lengths.…
Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234-avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ruškuc.
This short paper is concerned with the enumeration of permutations avoiding the following four patterns: $2431$, $4231$, $1432$ and $4132$. Using a bijective construction, we prove that these permutations are counted by the central binomial coefficients.
Given a real number beta>1, a permutation pi of length n is realized by the beta-shift if there is some x in [0,1] such that the relative order of the sequence x,f(x),...,f^{n-1}(x), where f(x) is the factional part of beta*x, is the same as that of the entries of pi. Widely studied from such diverse fields as number theory and automata theory, beta-shifts are prototypical examples one-dimensional chaotic dynamical systems. When beta is an integer, permutations realized by shifts where studied in [SIAM J. Discrete Math. 23 (2009), 765-786]. In this paper we generalize some of the results to ar…
In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {\em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collabor…
In 1990, Lakshmibai and Sandhya published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern avoidance was used to characterize geometrical properties of Schubert varieties. Their results are very closely related to work of Haiman, Ryan and Wolper, but Lakshmibai-Sandhya were the first to use that language exactly. Pattern avoidance in permutations was used historically by Knuth, Pratt, Tarjan, and others in the 1960's and 1970's to characterize sorting algorithms in computer science. Lascoux and Sch$\text{…
We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,...,n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the…
On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the \emph{Algebraic Eraser} scheme for key agreement over an insecure channel, using a novel hybrid of infinite and finite noncommutative groups. They also introduced the \emph{Colored Burau Key Agreement Protocol (CBKAP)}, a concrete realization of this scheme. We present general, efficient heuristic algorithms, which extract the shared key out of the public information provided by CBKAP. These algorithms are, according to heuristic reasoning and according to massive experiments, successful for all sizes of the security parameter…
A permutation p is realized by the shift on N symbols if there is an infinite word on an N-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as p. The set of realized permutations is closed under consecutive pattern containment. Permutations that cannot be realized are called forbidden patterns. It was shown in [Amigo, Elizalde, Kennel, J. Combin. Theory Ser. A 115 (2008) 485-504] that the shortest forbidden patterns of the shift on N symbols have length N+2. In this paper we give a characterization of the set of permutations that are…
In 1996, my brilliant student John Noonan, discovered, and proved that there are 3(2n)!/(n(n+3)!(n-3)!) ways to line-up n people of different heights in such a way that out of the n(n-1)(n-2)/6 possible triples of people exactly one is such that the tallest stands (not necessarily immediately) in front of the second-tallest, who in turn, stands (not necessarily immediately) in front of the shortest. In that article, I promised a prize of 25 dollars for a nice combinatorial proof. Alex Burstein gave such a proof. On Oct. 14, 2011, I talked about Alex's lovely proof at the Howard U. math colloqu…
This is a short survey about the theory of stable polynomials and its applications. It gives self-contained proofs of two theorems of Schrijver. One of them asserts that for a $d$--regular bipartite graph $G$ on $2n$ vertices, the number of perfect matchings, denoted by $\mathrm{pm}(G)$, satisfies $$\mathrm{pm}(G)\geq \bigg( \frac{(d-1)^{d-1}}{d^{d-2}} \bigg)^{n}.$$ The other theorem claims that for even $d$ the number of Eulerian orientations of a $d$--regular graph $G$ on $n$ vertices, denoted by $\varepsilon(G)$, satisfies $$\varepsilon(G)\geq \bigg(\frac{\binom{d}{d/2}}{2^{d/2}}\bigg)^n.$$…
We present a short proof of MacMahon's classic result that the number of permutations with $k$ inversions equals the number whose major index (sum of positions at which descents occur) is $k$
A short survey about combinatorics on words and algorithmic methods in a ring. Special attention is given to Shirshov's results. Adopted for undegraduate students.
We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time $n$, whose moves to the right or to the left are induced by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site $X$ at time $n$, obtain the probability measure of different excursions and define the asymptotic distribution…
After surveying some known properties of compact convex sets in the plane, we give a two rigorous proofs of the general feeling that supporting lines can be slide-turned slowly and continuously. Targeting a wide readership, our treatment is elementary on purpose.
We present a survey on Weil sums in which an additive character of a finite field $F$ is applied to a binomial whose individual terms (monomials) become permutations of $F$ when regarded as functions. Then we indicate how these Weil sums are used in applications, especially how they characterize the nonlinearity of power permutations and the correlation of linear recursive sequences over finite fields. In these applications, one is interested in the spectrum of Weil sum values that are obtained as the coefficients in the binomial are varied. We review the basic properties of such spectra, and …
