We define a statistic on the graph of commutation classes of a permutation of the symmetric group which is used to show that these graphs are equipped with a ranked poset structure, with a minimum and maximum. This characterization also allows us to compute the diameter of the commutation graph for any permutation, from which the results for the longest permutation and for fully commutative permutations are recovered.
We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.
We study three aspects of commutation classes of reduced decompositions: the number of commutation classes, the structures of their corresponding graphs, and the enumeration of subnetworks, a concept recently introduced by Warrington [21]. Our bound for the number of commutation classes generalizes the works of Knuth[12], Green and Losonczy [7], and Tenner [19]. We analyze the structure of the graph G(w) using pattern avoidance, which provides an application of Tenner's characterization of vexillary permutations in [19]. We also discuss some connections between our work and recent developments…
It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs …
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich in [KZ03] and correspond bijectively to the connected components of the strata of the moduli space of abelian differentials. In a paper [Boi14] Boissy proposed a Rauzy-type dynamics that acts on a s…
The Pancake graph($P_n$) represents the group of all permutations on n elements, namely $S_n$, with respect to the generating set containing all prefix reversals. The diameter of a graph is the maximum of all distances on the graph, where the distance between two vertices is the shortest path between them. In the case of the $P_n$, it is the maximum of the shortest generating sequence of each permutation in $S_n$. Here we propose a method to realise better upper bounds to the diameter of $P_n$ that has its focus on Graph Theoretical concepts rather than Algebra.
In this paper we introduce Yoke graphs, a family of flip graphs that generalizes several previously studied families of graphs: colored triangle free triangulations, arc permutations and caterpillars. Our main result is the computation of the diameter of an arbitrary Yoke graph.
General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $ \log |V| / \log |H| $ only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.
A permutation $\boldsymbol w$ gives rise to a graph $G_{\boldsymbol w}$; the vertices of $G_{\boldsymbol w}$ are the letters in the permutation and the edges of $G_{\boldsymbol w}$ are the inversions of $\boldsymbol w$. We find that the number of trees among permutation graphs with $n$ vertices is $2^{n-2}$ for $n\ge 2$. We then study $T_n$, a uniformly random tree from this set of trees. In particular, we study the number of vertices of a given degree in $T_n$, the maximum degree in $T_n$, the diameter of $T_n$, and the domination number of $T_n$. Denoting the number of degree-$k$ vertices in…
Let $Γ$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $Γ$ is bounded as $$\diam(Γ) \le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))},$$ where the maximization is over all permutations $π$, $c(π)$ denotes the number of cycles in $π$, and $\dist_T$ is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upp…
Given a finite group $G$ and a set $A$ of generators, the diameter diam$(Γ(G,A))$ of the Cayley graph $Γ(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam(G):= $\max_A$ diam$(Γ(G,A))$. It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$, but the best previously known upper bound was exponential in $\sqrt{n \log n}$. We give a quasipolynomial upper bound, namely, \[\text{diam}(G) = \exp(O((\log n)^4 \log\log n)) = \e…
Flip graphs are graphs on combinatorial objects in which the adjacency relation reflects a local change in the underlying objects. In this thesis we introduce Yoke graphs, a family of flip graphs that generalizes previously studied families of flip graphs on colored triangle-free triangulations, arc permutations and geometric caterpillars. Our main results are the computation of the diameter of an arbitrary Yoke graph and a full characterization of the automorphism group of this family of graphs. We also show that Yoke graphs are Schreier graphs of the affine Weyl group of type $\tilde{C}_m$. …
In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.
Given a generator set $S$ of the symmetric group ${\rm{Sym}}_n$, every permutation $π\in {\rm{Sym}_n}$ is a word (product of elements) of $S$. A positive integer $d(π)$ is associated with each $π\in{\rm{Sym}_n}$ taking the length of the shortest such word, and the $S$-diameter $d(S)$ is the maximum value of $d(π)$ with $π$ ranging over ${\rm{Sym_n}}$. The distance $d(π,ν)$ of two permutations $π,ν$ defined by $d(ν^{-1}\circπ)$ satisfies the axioms of a metric space. In this paper we consider the case where $S$ consists of all block transpositions of ${\rm{Sym_n}}$ and call $d(π)$ the block tra…
Let $G$ be a permutation group on a set $Ω$. A subset of $Ω$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $Σ(G)$, which we call the Saxl graph of $G$. The vertices of $Σ(G)$ are the points of $Ω$, and two vertices are adjacent if they form a base for $G$. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of $Σ(G)$ for a finite transitive group $G$, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For in…
The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter are called diametrically opposite vertices. The collection of shortest paths between diametrically opposite vertices are referred as diametral paths. In this work, we enumerate the number of diametral paths for Fibonacci cubes, Lucas cubes and Alternate Lucas cubes. We present bijective proofs that show that these numbers are related to alternating permutations and are enumerated by Euler numbers.
We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted IDENTIFYING CODE, (OPEN) LOCATING-DOMINATING SET and METRIC DIMENSION) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diame…
On the orbital diameter of classical groups in standard action
Hun-Ren Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；University of Manchester（曼彻斯特大学）；Heilbronn Institute for Mathematical Research（海布龙数学研究所）
A major problem in the study of combinatorial aspects of permutation groups is to determine the distances in the symmetric group $\Sym_n$ with respect to a generator set. One well-known such a case is when the generator set $S_n$ consists of block transpositions. It should be noted that "the block transposition distance of a permutation" is the distance of the permutation from the identity permutation in the Cayley graph $\Cay$, and "sorting a permutation by block transpositions" is equivalent to finding shortest paths in $\Cay$. The original results in our thesis concern the lower and upper b…
Let $g$, $h$ be a random pair of generators of $G=Sym(n)$ or $G=Alt(n)$. We show that, with probability tending to $1$ as $n\to \infty$, (a) the diameter of $G$ with respect to $S = \{g,h,g^{-1},h^{-1}\}$ is at most $O(n^2 (\log n)^c)$, and (b) the mixing time of $G$ with respect to $S$ is at most $O(n^3 (\log n)^c)$. (Both $c$ and the implied constants are absolute.) These bounds are far lower than the strongest worst-case bounds known (in Helfgott--Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap. Our r…
We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the automorphism group. The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. In 2014 Babai proved that the automorphism group of a strongly regular graph with $n$ vertices has minimal degree $\geq c n$, with known exceptions. St…
A problem of practical and theoretical interest is to determine or estimate the diameter of various families of Cayley networks. The previously known estimate for the diameter of Cayley graphs generated by transposition trees is an upper bound given in the oft-cited paper of Akers and Krishnamurthy (1989). In this work, we first assess the performance of their upper bound. We show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Evaluating their upper bound takes time $Ω(n!)$. In this paper,…
Our first main result is that correlations between monomers in the dimer model in $\mathbb{Z}^d$ do not decay to zero when $d > 2$. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas.…
Transactional data may be represented as a bipartite graph $G:=(L \cup R, E)$, where $L$ denotes agents, $R$ denotes objects visible to many agents, and an edge in $E$ denotes an interaction between an agent and an object. Unsupervised learning seeks to detect block structures in the adjacency matrix $Z$ between $L$ and $R$, thus grouping together sets of agents with similar object interactions. New results on quasirandom permutations suggest a non-parametric \textbf{four point test} to measure the amount of block structure in $G$, with respect to vertex orderings on $L$ and $R$. Take disjoint…
An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. The $n$-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The $n$-star gra…
The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmetric group $\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\cg(\sym(X))$…
CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)
Institut Curie, CNRS UMR168, Paris, France（居里研究所，法国国家科学研究中心UMR168，法国巴黎）；University of Washington（华盛顿大学）；Three Gorges Mathematical Research Center, China Three Gorges University（三峡数学研究中心，三峡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）；Independent Researcher（独立研究员）；Moscow State University（莫斯科国立大学）；Kazakh-British Technical University（哈萨克-英国技术大学）；Neapolis University Pafos, Cyprus（尼阿波利斯大学帕福斯校区，塞浦路斯）；City St George's, University of London（伦敦大学城市圣乔治学院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；ITMO University（ITMO大学）；Moscow Center for Advanced Studies（莫斯科高等研究中心）；National Research University Higher School of Economics（国立研究大学高等经济学院）；Ariel University（阿里埃勒大学）；Saint-Petersburg Pasteur Institute（圣彼得堡巴斯德研究所）；Criteo AI Lab, Berlin, Germany（Criteo人工智能实验室，德国柏林）；University of Bologna（博洛尼亚大学）；Faculty of Bioengineering and Bioinformatics, Lomonosov Moscow State University（莫斯科国立大学生物工程与生物信息学系）；Researcher at Banner Stat（Banner Stat研究员）；MIREA — Russian Technological University（MIREA — 俄罗斯技术大学）；Research Center of the Artificial Intelligence Institute, Innopolis University（因诺波利斯大学人工智能研究所研究中心）；Inpdependent Researcher（独立研究员）；LIX, Ecole Polytechnique/CNRS, IP Paris（LIX，巴黎综合理工学院/法国国家科学研究中心，巴黎理工学院）
