Chung and Graham define quasirandom subsets of $\mathbb{Z}_n$ to be those with any one of a large collection of equivalent random-like properties. We weaken their definition and call a subset of $\mathbb{Z}_n$ $ε$-balanced if its discrepancy on each interval is bounded by $εn$. A quasirandom permutation, then, is one which maps each interval to a highly balanced set. In the spirit of previous studies of quasirandomness, we exhibit several random-like properties which are equivalent to this one, including the property of containing (approximately) the expected number of subsequences of each order-type. We provide a few applications of these results, present a construction for a family of strongly quasirandom permutations, and prove that this construction is essentially optimal, using a result of W. Schmidt on the discrepancy of sequences of real numbers.
Previously, the author introduced quasirandom permutations, permutations of $\mathbb{Z}_n$ which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly so. Quasirandomness is established via discrete Fourier analysis and the Erdos-Turan inequality, as well as by other means. We apply our results on Sos permutations to make progress on a number of questions relating to the sequence of fractional parts of multiples of an irrational. Several intriguing new open problems are presented throughout the discussion…
For permutations P and T of lengths |P|\le|T|, let t(P,T) be the probability that the restriction of T to a random |P|-point set is (order) isomorphic to P. We show that every sequence \{T_j\} of permutations such that |T_j|\to\infty and t(P,T_j)\to 1/4! for every 4-point permutation P is quasirandom (that is, t(P,T_j)\to 1/|P|! for every P). This answers a question posed by Graham.
It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.
A set $S$ of permutations is forcing if for any sequence $\{Π_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(π,Π_i)$ converges to $\frac{1}{|π|!}$ for every permutation $π\in S$, it holds that $\{Π_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer $k$ such that the set of all permutations of order $k$ is forcing; this has been shown to be true for any $k\ge 4$. In particular, the set of all twenty-four permutations of order $4$ is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing se…
Let $G$ be a finite $D$-quasirandom group and $A \subset G^{k}$ a $δ$-dense subset. Then the density of the set of side lengths $g$ of corners \[ \{(a_{1},\dots,a_{k}),(ga_{1},a_{2},\dots,a_{k}),\dots,(ga_{1},\dots,ga_{k})\} \subset A \] converges to $1$ as $D\to\infty$.
We investigate extremal problems for quasirandom hypergraphs. We say that a $3$-uniform hypergraph $H=(V,E)$ is $(d,η)$-quasirandom if for any subset $X\subseteq V$ and every set of pairs $P\subseteq V\times V$ the number of pairs $(x,(y,z))\in X\times P$ with $\{x,y,z\}$ being a hyperedge of $H$ is in the interval $d|X||P|\pmη|V|^3$. We show that for any $\varepsilon>0$ there exists $η>0$ such that every sufficiently large $(1/2+\varepsilon,η)$-quasirandom hypergraph contains a tetrahedron, i.e., four vertices spanning all four hyperedges. A known random construction shows that the density $1…
Babai and Sós have asked whether there exists a constant c>0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy=z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.
Let $G$ be a finite group acting transitively on a set $Ω$. We study what it means for this action to be {\it quasirandom}, thereby generalizing Gowers' study of quasirandomness in groups. We connect this notion of quasirandomness to an upper bound for the convolution of functions associated with the action of $G$ on $Ω$. This convolution bound allows us to give sufficient conditions such that sets $S,T\subset G$ and $Γ\subseteq Ω$ contain elements $s\in S, t\in T, γ\inΓ$ such that $s(γ)=t$. Other consequences include an analogue of `the Gowers trick' of Nikolov and Pyber for general group act…
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $σ$ is the order type of the restriction of $σ: [n] \to [n]$ to a subset $S \subset [n]$. First, is it possible for the pattern counts in a permutation to be exactly equal to their expected values under a uniform distribution? Attempts to address this question lead naturally to an interesting number theoretic problem: when does $k!$ divide $\binom{n}{k}$? Second, if the tensor product of a permutation with large random permutations is random-lik…
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-…
For a fixed irrational number $α$ and $n\in \mathbb{N}$, we look at the shape of the sequence $(f(1),\ldots,f(n))$ after Schensted insertion, where $f(i) = αi \mod 1$. Our primary result is that the boundary of the Schensted shape is approximated by a piecewise linear function with at most two slopes. This piecewise linear function is explicitly described in terms of the continued fraction expansion for $α$. Our results generalize those of Boyd and Steele, who studied longest monotone subsequences. Our proofs are based on a careful analysis of monotone sets in two-dimensional lattices.
Define $S_n^k(T)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid all patterns in $T \subseteq S_m$. We enumerate $S_n^k(T)$, $T \subseteq S_3$, for all $|T| \geq 2$ and $0 \leq k \leq n$.
We show that the number of members of S_n avoiding any one of five specific triples of 4-letter patterns is given by sequence A111279 in OEIS, which is known to count weak sorting permutations. By numerical evidence, there are no other (non-trivial) triples of 4-letter patterns giving rise to this sequence. We make use of a variety of methods in proving our result, including recurrences, the kernel method, direct counting, and bijections.
Given a subset $S\subseteq\mathbb{P}$, let $\Pa(S;n)$ be the number of permutations in the symmetric group of ${1,2,...,n}$ that have peak set $S$. We prove a recent conjecture due to Billey, Burdzy and Sagan, which determines the sets that maximize $\Pa(S;n)$, where $S$ ranges over all subsets of ${1,2,...,n}$.
Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We construct a permutation of the integers avoiding arithmetic progressions of length $6$. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length $4$, sharpening the lower bound of $\frac{1}{3}$ from (LeSaulnier and Vijay, 2011). In addition, we generalize several…
Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$ and all finite subsets ${\mathcal X}\subset {\mathcal P}(V)$ such that every permutation on ${\mathcal X}$ can be extended to an element of ${\rm PGL}(V)$. Also, we reformulate the results in terms of linear and projective representations of symmetric groups.
We study properties of 3-restricted min-wise independent sets of permutations and prove that the class of these sets is isomorphic to the class of subsets of vertices of multi-dimentional hypercube of certain size with certain pair-wise distances fixed.
We introduce the "Median Inverse Problem" for metric spaces. In particular, having a permutation $π$ in the symmetric group $S_n$ (endowed with the breakpoint distance), we study the set of all $k$-subsets $\{x_1,...,x_k\}\subset S_n$ for which $π$ is a breakpoint median. The set of all $k$-tuples $(x_1,...,x_k)$ with this property is called the $k$-median inverse of $π$. Finding an upper bound for the cardinality of this set, we provide an asymptotic upper bound for the probability that $π$ is a breakpoint median of $k$ permutations $ξ_1^{(n)},...,ξ_k^{(n)}$ chosen uniformly and independently…
Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are: (a) being atomic, i.e. not being decomposable as a union of two downward closed proper subsets, or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are: (1) words over a finite alphabet under the consecutive subword ordering; and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are chara…
Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations π, σin S there is a point i in {1,...,n} such that π(i)=σ(i). Deza and Frankl \cite{MR0439648} proved that if S a subset of S(n) is intersecting then |S| \leq (n-1)!. Further, Cameron and Ku \cite{MR2009400} show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result.
This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (spec…
We introduce generalised orbit algebras. The purpose here is to measure how some combinatorial properties can characterize the action of a group of permutations on the subsets. The similarity with orbit algebras is such that it took the author a long time to find a generalised orbit algebra not arising from a permutation group.
Let $G$ be a permutation group acting on a finite set $Ω$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $Ω_m$ of all size $m$ subsets of $Ω$ satisfies the trivial inequalities $|Ω_m|/|G|\leq |Ω_m/G|\leq |Ω_m|$. The paper offers improvements of the upper bound in terms of the minimal degree of $G$ or the minimal degree of some its subset with a small complement. Applications include asymptotic enumeration of point configurations in an affine space over a finite field, unlabeled graphs and hypergraphs. Finally, with references to known results of permutation gr…
A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.
A permutation sigma in Sn is a k-derangement if for any subset X = {a1, . . ., ak} \subseteq [n], {sigma(a1), . . ., sigma(ak)} is not equal to X. One can form the k-derangement graph on the set of permutations of Sn by connecting two permutations sigma and tau if sigma(tau)^-1 is a k-derangement. We characterize when such a graph is connected or Eulerian. For n an odd prime power, we determine the independence, clique and chromatic number of the 2-derangement graph.
A family of permutations (\mathcal{A} \subset S_{n}) is said to be (t)-\textit{intersecting} if any two permutations in (\mathcal{A}) agree on at least (t) points, i.e. for any (σ, π\in \mathcal{A}), (|\{i \in [n]: σ(i)=π(i)\}| \geq t). It was recently proved by Friedgut, Pilpel and the author that for (n) sufficiently large depending on (t), a (t)-intersecting family (\mathcal{A} \subset S_{n}) has size at most ((n-t)!), with equality only if (\mathcal{A}) is a coset of the stabilizer of (t) points (or `(t)-coset' for short), proving a conjecture of Deza and Frankl. Here, we first obtain a ro…
A permutation of n letters is k-prolific if each (n-k)-subset of the letters in its one-line notation forms a unique pattern. We present a complete characterization of k-prolific permutations for each k, proving that k-prolific permutations of m letters exist for every m \ge k^2/2+2k+1, and that none exist of smaller size. Key to these results is a natural bijection between k-prolific permutations and certain "permuted" packings of diamonds.
Given a permutation statistic $s : S_n \to \mathbb{R}$, define the mean statistic $\bar{s}$ as the statistic which computes the mean of $s$ over conjugacy classes. We describe a way to calculate the expected value of $s$ on a product of $t$ independently chosen elements from the uniform distribution on a union of conjugacy classes $Γ\subseteq S_n$. In order to apply the formula, one needs to express the class function $\bar{s}$ as a linear combination of irreducible $S_n$-characters. We provide such expressions for several commonly studied permutation statistics, including the excedance number…
Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.
In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over $\mathbb{F}_{q^2}$, in the context of rank deficient and full rank linear maps over $\mathbb{F}_{q^2}$. We derive necessary and sufficient conditions under which the given class of polynomials are permutation polynomials. We further show that the number of such permutation polynomials can be easily enumerated. Only a subset of these permutation polynomials have been reported …
Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $σ,π\in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset \operatorname{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\leq n$ and $\mathcal{F} \subset \operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\leq t!(n-t)!$ and equality holds only if $\ma…
A family of permutations $\mathcal{F} \subset S_{n}$ is said to be $t$-intersecting if any two permutations in $\mathcal{F}$ agree on at least $t$ points. It is said to be $(t-1)$-intersection-free if no two permutations in $\mathcal{F}$ agree on exactly $t-1$ points. If $S,T \subset \{1,2,\ldots,n\}$ with $|S|=|T|$, and $π: S \to T$ is a bijection, the $π$-star in $S_n$ is the family of all permutations in $S_n$ that agree with $π$ on all of $S$. An $s$-star is a $π$-star such that $π$ is a bijection between sets of size $s$. Friedgut and Pilpel, and independently the first author, showed tha…
Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least $3$. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O\times O$ under the diagonal action of the automorphism group.
The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patte…
We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions enumerating these two statistics. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique is to use bijections between pattern-avoiding permutati…
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 si…
The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.
Given a permutation $f$, we study the positroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$-polynomials coincide with the generalized $q,t$-Catalan numbers that recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov-Rozansky hom…
We look at geometric limits of large random non-uniform permutations. We mainly consider two theories for limits of permutations: permuton limits, introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion of scaling limits for permutations; and Benjamini-Schramm limits, introduced by the author to define a notion of local limits for permutations. The models of random permutations that we consider are mainly constrained models, that is, uniform permutations belonging to a given subset of the set of all permutations. We often identify this subset using pattern-avoidance, fo…
A set of $N$ permutations of $\{1,2,\dots,v\}$ is $(N,v,t)$-suitable if each symbol precedes each subset of $t-1$ others in at least one permutation. The central problems are to determine the smallest $N$ for which such a set exists for given $v$ and $t$, and to determine the largest $v$ for which such a set exists for given $N$ and $t$. These extremal problems were the subject of classical studies by Dushnik in 1950 and Spencer in 1971. We give examples of suitable sets of permutations for new parameter triples $(N,v,t)$. We relate certain suitable sets of permutations with parameter $t$ to o…
An (n,d)-permutation code is a subset C of Sym(n) such that the Hamming distance d_H between any two distinct elements of C is at least equal to d. In this paper, we use the characterisation of the isometry group of the metric space (Sym(n),d_H) in order to develop generating algorithms with rejection of isomorphic objects. To classify the (n,d)-permutation codes up to isometry, we construct invariants and study their efficiency. We give the numbers of non-isometric (4,3)- and (5,4)- permutation codes. Maximal and balanced (n,d)-permutation codes are enumerated in a constructive way.
Selecting N random points in a unit square corresponds to selecting a random permutation. By putting 5 types of symmetry restrictions on the points, we obtain subsets of permutations : involutions, signed permutations and signed involutions. We are interested in the statistics of the length of the longest up/right path of random points selections in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of Painlevé II equation. Some of them are Tracy-Widom distributions in random matrix theory, while there are two new classe…
A restricted permutation of a locally finite directed graph $G=(V,E)$ is a vertex permutation $π: V\to V$ for which $(v,π(v))\in E$, for any vertex $v\in V$. The set of such permutations, denoted by $Ω(G)$, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser (2016) of restricted $\mathbb{Z}^d$ permutations, in which $Ω(G)$ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this corres…
