We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc. with certain minimality and transitivity conditions imposed on the factors. The method is to encode such factorizations as planar maps with certain "descent structure" and apply a new combinatorial decomposition to make their enumeration more manageable. We apply our technique to count factorizations of permutations with one or two parts, extending earlier work of Goulden and Jackson. We also show how these methods are readily modified to count inequivalent factorizations, where equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits a substantial generalization of recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of their analysis.
We give two combinatorial proofs of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type $B$.
Constant composition codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication. In particular, a certain class of constant composition codes called frequency permutation arrays have been suggested as ideal, in some sense, for these purposes. In this paper we characterise a family of neighbour transitive codes in Hamming graphs in which frequency permutation arrays play a central rode. We also classify all the permutation codes generated by groups in this family.
We consider an experimental design problem for permutations: given a fixed set $X$, and an integer $t$, construct a list $L$ of permutations of $X$ such that every ordered $t$-tuple of distinct elements of $X$ occurs as a consecutive subsequence of exactly one permutation in $L$. In this paper we focus on solutions based on sharply transitive group actions, in effect generalizing Gordon's notion of group sequencing. We give an explicit construction when $|X|$ is prime for the case $t=3$, and analyze a branching algorithm for the general case which produces, for example, a rare design with $t=6…
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.
For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in \mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $\mathcal{F}$ runs over all intersecting sets in $G$. If $G_v$ is a largest intersecting set in $G$ then $G$ is said to have the Erdős-Ko-Rado (EKR)-property. This paper is devoted to the study of transitive…
Given two positive integers n and c, we determine an upper bound, as a function of n and c, for the maximum order of a finite nilpotent transitive group of degree n and nilpotency class at most c.
When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that existence of automorphisms acting as even permutations on the vertex set of a graph, called {\em even automorphisms}, forces existence of automorphisms that act as odd permutations, called {\em odd automorphisms}. As a first step towards resolving the above question, a complete information on existence of odd automorphisms in cubi…
The number of minimal transitive star factorizations of a permutation was shown by Irving and Rattan to depend only on the conjugacy class of the permutation, a surprising result given that the pivot plays a very particular role in such factorizations. Here, we explain this symmetry and provide a bijection between minimal transitive star factorizations of a permutation πhaving pivot k and those having pivot k'.
In this paper, we introduce a new parameter of a code, referred to as the remoteness, which can be viewed as a dual to the covering radius. Indeed, the remoteness is the minimum radius needed for a single ball to cover all codewords. After giving some general results about the remoteness, we then focus on the remoteness of permutation codes. We first derive upper and lower bounds on the minimum cardinality of a code with a given remoteness. We then study the remoteness of permutation groups. We show that the remoteness of transitive groups can only take two values, and we determine the remoten…
We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.
In the first paper in this series we estimated the probability that a random permutation $π\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $π$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\dots,k_m$, where $k_1+\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estim…
The minimal degree of a permutation group $G$ is defined as the minimal number of non-fixed points of a non-trivial element of $G$. In this paper we show that if $G$ is a transitive permutation group of degree $n$ having no non-trivial normal $2$-subgroups such that the stabiliser of a point is a $2$-group, then the minimal degree of $G$ is at least $\frac{2}{3}n$. The proof depends on the classification of finite simple groups.
We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form $(1 i)$. Our result generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored.
Two factorizations of a permutation into products of cycles are equivalent if one can be obtained from the other by repeatedly interchanging adjacent disjoint factors. This paper studies the enumeration of equivalence classes under this relation. We obtain closed form expressions for generating series of inequivalent minimal transitive factorizations of permutations having up to three cycles, and also of permutations with four cycles when factors are restricted to be transpositions. Our derivations rely on a new correspondence between inequivalent factorizations and acyclic alternating digraph…
The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of $2$-neighbour-transitive codes than previous classification results of families of $2$-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Sev…
We consider the problem of factoring permutations as a product of special types of transpositions, namely, those transpositions involving two positions with bounded distances. In particular, we investigate the minimum number, $δ$, such that every permutation can be factored into no more than $δ$ special transpositions. This study is related to sorting algorithms, Cayley graphs, and genomics.
Let $A_2(n,d)$ be the maximum size of a binary code of length $n$ and minimum distance $d$. In this paper we present the following new lower bounds: $A_2(18,4) \ge 5632$, $A_2(21,4) \ge 40960$, $A_2(22,4) \ge 81920$, $A_2(23,4) \ge 163840$, $A_2(24,4) \ge 327680$, $A_2(24,10) \ge 136$, and $A_2(25,6) \ge 17920$. The new lower bounds are a result of a systematic computer search over transitive permutation groups.
We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.
We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley grap…
Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where ${\cal F}$ runs over all intersecting sets in $G$ and $G_v$ is a stabilizer of $v\in V$. In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be eit…
A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type $[L_1,L_2]$ and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally $[L_1,L_2]$ graphs. We also consider the corresponding question for amalgams of rank $k\geq 3$. We completely resolve this by showing that the order of the Bo…
The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected $2$-arc-transitive graphs of a fixed valence tends to $0$ as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-$L$ graphs, …
In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and…
Let $Ω=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $Ω$ is the integer partition $λ=(λ_1,...,λ_k)$ defined by $λ_i = |P_i|$. Let G be a group of permutations acting on $Ω$. For a fixed partition $λ$ of n, we say that G is {\em $λ$-transitive} if G has only one orbit when acting on partitions P of shape $\la$. A corresponding definition can also be given when G is just a set. For example, if $λ=(n-t,1,...,1)$, then a $λ$-transitive group is the same as a t-transitive permutation group and if $λ=(n-t,t)$, then we recover the t-homogeneous perm…
Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $ρ(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and ${\cal F}$ runs over all intersecting sets in $G$. Intersection densities of transitive groups of degree $pq$, where $p>q$ are odd primes, is considered. In particular, t…
Let $Γ$ be a connected $G$-arc-transitive graph, let $uv$ be an arc of $Γ$ and let $L$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. We study the problem of bounding $|G_{uv}|$ in terms of $L$ and the order of $Γ$.
Let $p$ be a prime and let $L$ be either the intransitive permutation group $C_p\times C_p$ of degree $2p$ or the transitive permutation group $C_p \wr C_2$ of degree $2p$. Let $Γ$ be a connected $G$-vertex-transitive and $G$-edge-transitive graph and let $v$ be a vertex of $Γ$. We show that if the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is isomorphic to $L$ then either $|V(Γ)|\geq p|G_v|\log_p\left(|G_v|/2\right)$, or $|V(Γ)|$ is bounded by a constant depending only on $p$, or $Γ$ is a very-well understood graph. This generalises a few recent resul…
Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share certain elementary combinatorial features are related by conjugations as above, and dessins d'enfants in the same Galois-…
In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(Ω)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is either 2-transitive or a Frobenius group, then $|{S}|\leqslant|G_ω|$ (for some $ω\in Ω$). Furthermore, for some 2-transitive groups, $|{S}|=|G_ω|$ if and only if ${S}$ is a coset of a stabilizer. In this paper, we prove that these statements are far from the truth for general transitive groups. In particular, we show that in the case of primitive groups, there is eve…
It is known that there are precisely three transitive permutation groups of degree $6$ that admit an invariant partition with three parts of size $2$ such that the kernel of the action on the parts has order $4$; these groups are called $A_4(6)$, $S_4(6d)$ and $S_4(6c)$. For each $L\in \{A_4(6), S_4(6d), S_4(6c)\}$, we construct an infinite family of finite connected $6$-valent graphs $\{Γ_n\}_{n\in \mathbb{N}}$ and arc-transitive groups $G_n \le \rm{Aut}(Γ_n)$ such that the permutation group induced by the action of the vertex-stabiliser $(G_n)_v$ on the neighbourhood of a vertex $v$ is permu…
Let $\M(A,θ)$ be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet $B\subset A$ such that the right factor of a bisection $\M(A,θ)=\M(B,θ\_B).T$ be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of $\M(A,θ)$ and associated bases of $L\_K(A,θ)$.
Let $Γ$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $Γ$ and let $L=G_v^{Γ(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. Then $(Γ,G)$ is said to be \emph{locally-$L$}. A transitive permutation group $L$ is \emph{graph-restrictive} if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(Γ,G)$ and an arc $(u,v)$ of $Γ$, the inequality $|G_{uv}|\leq c(L)$ holds. Using this terminology, the Weiss Conjecture says that primitive groups are graph-restrictive. We propose a very strong generalisati…
A factorization of a permutation into transpositions is called "primitive" if its factors are weakly ordered. We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described.
We develop the relationship between minimal transitive star factorizations and noncrossing partitions. This gives a new combinatorial proof of a result by Irving and Rattan, and a specialization of a result of Kreweras. It also arises in a poset on the symmetric group whose definition is motivated by the Subword Property of the Bruhat order.
Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.
We consider a new type of Hurwitz number, the number of ordered transitive factorizations of an arbitrary permutation into d-cycles. In this paper, we focus on the special case d = 3. The minimal number of transitive factorizations of any permutation into 3-cycles has been worked out by David, Goulden and Jackson. Also, such factorizations for transpositions, the case d = 2, have been considered by Crescimanno and Taylor. Goulden and Jackson have proved the differential equation for the generating series of simple Hurwitz numbers. Based on their results, we use W-operator to prove a differenti…
An automorphism group G of a 1-factorization of the complete multipartite graph $K_{m\times n}$ consists in permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence or non-existence problem of a 1-factorization of $K_{m\times n}$ admitting an abelian group acting sharply transitively on the vertices of the graph.
We prove that the Hurwitz action on reflection factorizations of Coxeter elements is transitive up to certain natural constraints in the complex reflection groups G4 and G5. This affirms a more general conjecture by Lewis and Reiner in these specific cases. The proof uses induction on length of the factorization using the fact that the square of a reflection is also a reflection.
We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
A subset of a group G of Sym(n) is intersecting if for any pair of permutations $π,σ\in G$ there is an $i$ in {1,2,...,n} such that $π(i) = σ(i)$. It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(n), Alt(n) and PGL(2,q) are exactly the cosets of the point-stabilizers. In this paper, we show how this method can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transtive groups with degree no more than 20.
Let $Γ$ be a finite $G$-vertex-transitive digraph. The in-local action of $(Γ,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_+$ is defined analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some general results, but pay special attention to the case when $L_-$ and $L_+$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the w…
Each group G of nxn permutation matrices has a corresponding permutation polytope, P(G):=conv(G) in R^{nxn}. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t,floor(n/2)} is a sharp upper bound on the diameter of the graph of P(G); so if G is transitive, the diameter is at most 2. We also show that P(G) achieves its maximal dimension of (n-1)^2 precisely when G is 2-transitive. We then extend results of I. Pak on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving…
Let $G$ be a permutation group on a finite set $Ω$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$ having the same orbits as $G$ on the $k$-th Cartesian power $Ω^k$ of $Ω$. A group $G$ is called $\frac{3}{2}$-transitive if its transitive and the orbits of a point stabilizer $G_α$ on the set $Ω\setminus\{α\}$ are of the same size greater than one. We prove that the $2$-closure $G^{(2)}$ of a $\frac{3}{2}$-transitive permutation group $G$ can be found in polynomial time in size of $Ω$. In addition, if the group $G$ is not $2$-transitive, then for ev…
A transitive group $G$ of permutations of a set $Ω$ is primitive if the only $G$-invariant equivalence relations on $Ω$ are the trivial and universal relations. If $α\in Ω$, then the orbits of the stabiliser $G_α$ on $Ω$ are called the $α$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $α$-suborbits are the subdegrees of $G$. If $G$ acts primitively on an infinite set $Ω$, and all the suborbits of $G$ are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of $G$ as a non-decreasing sequence $1 = m_0 \leq m_1 \leq ...$, the subdegree growth rates of …
In this paper, we consider finite flag-transitive affine planes with a solvable automorphism group. Under a mild number-theoretic condition involving the order and dimension of the plane, the translation complement must contain a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. We develop a new approach to the study of such planes by associating them with planar functions and permutation polynomials in the odd order and even order case respectively. In the odd order case, we characterize the Kantor-Suetake family by using Menichetti's clas…
Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,n}, are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the centre of S_n, and show that they have a polynomiality property. These centrality and polynomiality properti…
