pinecone/core-2020-05-10-deduplication huggingface dataset:
Transitive Factorizations in the Symmetric Group, and Combinatorial Aspects of Singularity Theory

AbstractWe consider the determination of the number ck(α) of ordered factorizations of an arbitrary permutation on n symbols, with cycle distribution α, intok -cycles such that the factorizations have minimal length and the group generated by the factors acts transitively on then symbols. The case k= 2 corresponds to the celebrated result of Hurwitz on the number of topologically distinct holomorphic functions on the 2-sphere that preserve a given number of elementary branch point singularities. In this case the monodromy group is the full symmetric group. For k= 3, the monodromy group is the alternating group, and this is another case that, in principle, is of considerable interest. We conjecture an explicit form, for arbitrary k, for the generating series for ck(α), and prove that it holds for factorizations of permutations with one, two and three cycles (so α is a partition with at most three parts). Our approach is to determine a differential equation for the generating series from a combinatorial analysis of the creation and annihilation of cycles in products under the minimality condition
Supersolvable Frame-matroid and Graphic-lift Lattices

AbstractA geometric lattice is a frame if its matroid, possibly after enlargement, has a basis such that every atom lies under a join of at most two basis elements. Examples include all subsets of a classical root system. Using the fact that finitary frame matroids are the bias matroids of biased graphs, we characterize modular coatoms in frames of finite rank and we describe explicitly the frames that are supersolvable. We apply the characterizations to three kinds of example. A geometric lattice is a graphic lift if it can be extended to contain an atom whose upper interval is graphic. We characterize modular coatoms in and supersolvability of graphic lifts of finite rank and we examine families analogous to the frame examples
Random Subgraphs of Cayley Graphs overp-Groups

AbstractThe subject of this paper is the size of the largest component in random subgraphs of Cayley graphs, Xn, taken over a class of p -groups, Gn. Gnconsists of p -groups, Gn, with the following properties: (i)Gn/Φ(Gn) ∼ =Fpn, where Φ(Gn) is the Frattini subgroup and (ii) | Gn| ≤nKn, where K is some positive constant. We consider Cayley graphs Xn=Γ(Gn, Sn′), where Sn′=Sn∪Sn−1, and Snis a minimal Gn-generating set. By selecting Gn-elements with the independent probability λnwe induce random subgraphs of Xn. Our main result is, that there exists a positive constant c> 0 such that for λn=c ln(| Sn′ |)/| Sn′ | the largest component of random induced subgraphs of Xncontains almost all vertices
Further Investigations Involving Rook Polynomials With Only Real Zeros

AbstractWe study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham . Another involves a version of the ‘hit polynomial’ of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is analogous to a theorem of the author, K. Ono, and D. G. Wagner, namely that for Ferrers boards the hit polynomial has only real zeros. We also show that for each of these families there is a general conjecture involving arrays of numbers satisfying inequalities which contains these theorems as special cases. We provide evidence for the truth of these conjectures by proving other special cases and discussing computational experiments
On Extending Closure Systems to Matroids

AbstractGeometric representations of data are of main interest in data analysis. Generalizing the idea of linear representations this paper is concerned with the representability of data by matroids which leads to the problem of extending closure systems to matroids. Finite matroid constructions are introduced in order to characterize the class of set systems which can be extended to the set of hyperplanes of a matroid. Since this constructive characterization of extendable set systems does not give simple conditions for extendability, the question arises of whether such simple conditions actually exist. By means of an infinite series of critical configurations and Łoś’s theorem from model theory it is proven that there is no finite set of first-order axioms characterizing the class of set systems that admit a matroid construction. These results are discussed and perspectives for further research are given
Nonexistence of some Antipodal Distance-regular Graphs of Diameter Four

AbstractWe find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Jurišić, Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier
Minors and Strong Products

AbstractLet G⊠Hand G□H denote, respectively, the strong and Cartesian products of graphs G and H. (We recall thatK2⊠K2 is the complete graph K4on four vertices, whileK2□K2 is a four-cycle C4.) Using a simple construction, we show that, for every bipartite G, the graph G⊠K2is a minor of the graphG□C4 . In particular, the d -cube Qdhas a complete minor on 2⊠(d+ 1)/2vertices if d is odd, and on 3 · 2⊠(d− 2)/2vertices if d is even. We do not know whether such a complete minor of Qdis largest possible
On a Family of Planes of a Polar Space

AbstractA Y -family S is a set of q+ 3 planes ofPG (5, q) satisfying some conditions on their intersections. If every member of S is isotropic for a form f on PG(5, q), the affine expansion of S is covered by a geometry insidePG (7, q) (Proposition ?? ). Such families are classified if f is unitary (Proposition ?? ). Existence of such families with f symplectic is equivalent to that of two o -polynomials with certain properties (Lemma ?? ). A nonexistence result follows from this criterion (Proposition ?? )
Choosability, Edge Choosability, and Total Choosability of Outerplane Graphs

AbstractLet χl(G),χl′ (G),χl′′ (G), and Δ(G) denote, respectively, the list chromatic number, the list chromatic index, the list total chromatic number, and the maximum degree of a non-trivial connected outerplane graph G. We prove the following results. (1) 2 ≤χl(G) ≤ 3 andχl (G) = 2 if and only if G is bipartite with at most one cycle. (2)Δ(G) ≤χl′(G) ≤Δ(G) + 1 andχl′ (G) =Δ(G) + 1 if and only if G is an odd cycle. This proves the well-known list edge coloring conjecture for outerplane graphs. (3)χl′′(G) =Δ(G) + 1 if Δ(G) ≥ 4 and χl′′(G) ≤ 5 if Δ(G) ≤ 3. This proves a conjecture of O. V. Borodin, A. V. Kostochka and D. R. Woodall, List edge and list total coloring of multigraphs, J. Comb. Theory B, 71 (1997), 184–204 for outerplane graphs
Regular Maps on Surfaces with Large Planar Width

AbstractA map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the flags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integersp≥ 3 and q≥ 3 satisfying 1/p+ 1/q≤ 1/2, there is an orientable regular map with face-size p and valency q such that every non-contractible simple closed curve on the surface meets the 1-skeleton of the map in at least w points. This result has several interesting consequences concerning maps on surfaces, graphs and related concepts. For example, MacBeath’s theorem about the existence of infinitely many Hurwitz groups, or Vince’s theorem about regular maps of given type (p, q), or residual finiteness of triangle groups, all follow from our result
A Characterization of Sturmian Words by Return Words

AbstractNous présentons une nouvelle caractérisation des mots de Sturm basée sur les mots de retour. Si l’on considère chaque occurrence d’un mot w dans un mot infini récurrent, on définit l’ensemble des mots de retour de w comme l’ensemble de tous les mots distincts commençant par une occurrence de w et finissant exactement avant l’occurrence suivante de w. Le résultat principal montre qu’un mot est sturmien si et seulement si pour chaque mot w non vide apparaissant dans la suite, la cardinalité de l’ensemble des mots de retour de w est égale à deux. We present a new characterization of Sturmian words using return words. Considering each occurrence of a word w in a recurrent word, we define the set of return words over w to be the set of all distinct words beginning with an occurrence of w and ending exactly before the next occurrence of w in the infinite word. It is shown that an infinite word is a Sturmian word if and only if for each non-empty word w appearing in the infinite word, the cardinality of the set of return words over w is equal to two
A Note on the Orlik–Solomon Algebra

AbstractLet M=M(E) be a matroid on a linear ordered set E. The Orlik–Solomon Z -algebra OS(M) of M is the free exteriorZ -algebra on E, modulo the ideal generated by the circuit boundaries. The Z -moduleOS (M) has a canonical basis called ‘no broken circuit basis’ and denotednbc. Let eX=∏ei,ei∈X⊂E. We prove that wheneX is expressed in thenbcbasis, then all the coefficients are 0 or ±1. We present here an algorithm for computing these coefficients. We prove in appendix a numerical identity involving the dimensions of the algebras of Orlik–Solomon of the minors of a matroid and its dual
On Unit Circles Which Avoid All but Two Points of a Given Point-set

AbstractIn this paper we show that if a finite set of at least two points in the plane has a diameter less than 3, then there is a unit circle passing through exactly two points of the set. I conjecture that if the diameter of the point-set is between 3 and 2, then the same statement is true with only one exception. The exceptional set must have four points such that three of the points are vertices of an acute triangle with circum-radius 1, and the fourth point is the common point of the three unit circles which go through only two vertices of this triangle
Six Points Suffice: How to Check for Metric Consistency

AbstractIn many areas of data analysis, it is desirable to have tools at hand for analyzing the structure of distance tables—or, in more mathematical terms, of finite metric spaces. One such tool, known as split decomposition theory has proven particularly useful in this respect. The class of so-called totally decomposable metrics forms a cornerstone for this theory, and much work has been devoted to their study. Recently, it has become apparent that a particular subclass of these metrics, the consistent metrics, are also of fundamental importance. In this paper, we give a six-point characterization of consistent metrics amongst the totally decomposable ones
Caps on Classical Varieties and their Projections

AbstractA family of caps constructed by G. L. Ebert, K. Metsch and T. Szönyi results from projecting a Veronesian or a Grassmannian to a suitable lower-dimensional space. We improve on this construction by projecting to a space of much smaller dimension. More precisely, we partition PG(3 r− 1, q) into a (2 r− 1)-space, an (r− 1)-space andqr− 1 cyclic caps, each of size (q2r− 1)/(q− 1). We also decide when one of our caps can be extended by a point from the (2 r− 1)-space or the (r− 1)-space. The proof of the results uses several ingredients, most notably hyperelliptic curves
Orbits onn-tuples for Infinite Permutation Groups

AbstractThis paper presents a theorem on the growth rate of the orbit-counting sequences of a primitive oligomorphic group: if G is not a highly homogeneous group, then the growth rate for the sequence counting orbits onn -tuples of distinct elements is bounded below by cnn!, wherec≈ 1.172. The previously known lower bounds concerned all not highly transitive groups, including highly homogeneous groups which are known to have roughly factorial growth rate. This paper shows that highly homogeneous groups are the only groups with such a growth rate, while for all other primitive groups the growth rate is faster and the bound is improved by an exponential factor
Complete Rotations in Cayley Graphs

AbstractAs it is introduced by Bermond, Pérennes, and Kodate and by Fragopoulou and Akl, some Cayley graphs, including most popular models for interconnection networks, admit a special automorphism, called complete rotation. Such an automorphism is often used to derive algorithms or properties of the underlying graph. For example, some optimal gossiping algorithms can be easily designed by using a complete rotation, and the constructions of the best known edge disjoint spanning trees in the toroidal meshes and the hypercubes are based on such an automorphism. Our purpose is to investigate such Cayley graphs. We relate some symmetries of a graph with potential algebraic symmetries appearing in its definition as a Cayley graph on a group
Spin Models, Association Schemes and the Nakanishi–Montesinos Conjecture

AbstractA 3-transformation of a link is a local change which replaces two strings that are three times half twisted around each other by two untwisted strings (and vice versa). The Nakanishi–Montesinos (NM) conjecture asserts that this 3-transformation can unknot any link. We introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformation. We try to classify such spin models and determine the corresponding link invariant. It is proved that the dimension of the Bose–Mesner algebra generated by the spin model is ≤4. For dimension 1 and 2, there is no such spin model, but for dimension 3, there exists a unique one. Its link invariant is a non-trivial specialization of the Kauffman polynomial, but does not distinguish trivial links from the others, and hence cannot disprove the NM conjecture. For dimension 4, we give a family of NM-spin models. The corresponding link invariant is identified and does not distinguish trivial links from the others. Strong regularity and triple regularity of the Bose–Mesner algebra and its fusions are studied
A Cyclage Poset Structure for Littlewood–Richardson Tableaux

AbstractA graded poset structure is defined for sets of Littlewood–Richardson (LR) tableaux whose cardinalities are the multiplicities of irreducible gl(n)-modules in the tensor product of several irreducible gl(n)-modules indexed by rectangular partitions. This is a generalization of the cyclage poset on tableaux defined by Lascoux and Schützenberger. This combinatorial construction is of independent interest and may be understood without reference to the underlying algebraic geometry. It is shown that the polynomials obtained by enumerating LR tableaux by shape and a generalized charge statistic, are the graded multiplicities of irreducibles in certain graded gl(n)-modules supported in the closure of a nilpotent conjugacy class. In particular, explicit tableau formulas are obtained for the special cases of the Kostka–Foulkes polynomials, the coefficient polynomials of two-column Macdonald–Kostka polynomials, and the graded multiplicities of coordinate rings of closures of conjugacy classes of nilpotent matrices. These polynomials coincide with the q -enumeration of rigged configurations and conjecturally coincide with the q -analogues of LR coefficients defined by the spin–weight generating functions of ribbon tableaux introduced by Lascoux, Leclerc, and Thibon
Multi-atoms and Monotonicity of Generalized Kostka Polynomials

AbstractThere is a certain family of Poincaré polynomials that arise naturally in geometry. They satisfy a monotonicity property and admit a combinatorial description in terms of a graded poset whose elements are called Littlewood–Richardson tableaux. The purpose of this article is to give a combinatorial explanation of the monotonicity by exhibiting embeddings of these graded posets. In particular, these posets embed into the cyclage poset of Lascoux and Schützenberger, which was introduced to explain the monotonicity of the Kostka–Foulkes polynomials. The image of the embedding into the cyclage poset is characterized, giving another combinatorial description of the Poincaré polynomials in terms of the catabolizable tableaux defined by the author and J. Weyman
Sets with few Intersection Numbers from Singer Subgroup Orbits

AbstractUsing a Singer cycle in Desarguesian planes of order q≡ 1(mod3), q a prime power, Brouwer 2 gave a construction of sets such that every line of the plane meets them in one of three possible intersection sizes. These intersection sizes x, y, and z satisfy the system of equations[formula]Brouwer claimed that this system has a unique solution in integers. Further, Brouwer noted that for q a perfect square, this system has a solution for which two of the variables are equal, ostensibly implying that when q is a square the constructed set has only two intersection numbers. In this paper, we perform a detailed analysis which shows that this system does not in general have a unique solution. In particular, the constructed sets when q is a square might have three intersection numbers. The cases for which this occurs are completely determined
Semisymmetry of Generalized Folkman Graphs

AbstractA regular edge- but not vertex-transitive graph is said to be semisymmetric. The study of semisymmetric graphs was initiated by Folkman, who, among others, gave constructions of several infinite families such graphs . In this paper a generalization of his construction for orders a multiple of 4 is proposed, giving rise to some new families of semisymmetric graphs. In particular, one associated with the cyclic group of order n, n≥ 5, which belongs to the class of tetracirculants, that is, graphs admitting an automorphism with precisely four orbits, all of the same length. Semisymmetry properties of tetracirculants are investigated in greater detail, leading to a classification of all semisymmetric graphs of order 4 p, where p is a prime
The Penultimate Rate of Growth for Graph Properties

AbstractGiven a property P of graphs, write Pnfor the set of graphs with vertex set [ n ] having property P. We call | Pn| the speed of P. Recent research has shown that the speed of a monotone or hereditary property P can be a constant, polynomial, or exponential function of n, and the structure of the graphs in P can then be well described. Similarly, | Pn| can be of the form n(1−1/k+o(1))nor 2(1−1/k+o(1))n2/2for some positive integer k> 1 and the properties can be described and have well-behaved speeds. In this paper, we discuss the behavior of properties with speeds between these latter bounds, i.e., between n(1+o(1))nand 2(1/2+o(1))n2/2
Non-constructible Complexes and the Bridge Index

AbstractWe show that if a three-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres. We also obtain similar bounds concluding that a 3-sphere or 3-ball is non-shellable or not vertex decomposable. These two last bounds are sharp
On a Mutation Problem for Oriented Matroids

AbstractFor uniform oriented matroids M with n elements, there is in the realizable case a sharp lower bound Lr(n) for the number mut(M) of mutations of M: Lr(n) =n≤mut(M), see Shannon [17]. Finding a sharp lower bound L(n) ≤mut(M) in the non-realizable case is an open problem for rank d≥ 4. Las Vergnas [9] conjectured that 1 ≤L(n). We study in this article the rank 4 case. Richter-Gebert [11] showed thatL (4 k) ≤ 3 k+ 1 for k≥ 2. We confirm Las Vergnas’ conjecture for n< 13. We show that L(7k+c) ≤ 5 k+c for all integersk≥ 0 and c≥ 4, and we provide a 17 element example with a mutation free element
Oriented Lagrangian Matroids

AbstractIn this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components. The definitions are made first in terms of a combinatorial axiomatisation, and then again in terms of elementary geometric properties of the Coxeter matroid polytope. We also generalise the concept of rank and signature of a quadratic form to symplectic Lagrangian matroids in a surprisingly natural way
Multimatroids III. Tightness and Fundamental Graphs

AbstractThis paper continues the study of multimatroids. Here we introduce the subclass of tight multimatroids, which contains the liftings of even delta-matroids, the 3-matroids derived from isotropic systems, the Eulerian 3-matroids associated to 4-regular graphs and the Eulerian 2-matroids associated to evenly directed 4-regular graphs. The local properties of a tight multimatroid in the vicinity of a base are reflected by a fundamental graph, as in matroid theory. We describe how the fundamental graph is transformed when the base is modified. As an application we derive some connectivity properties of tight multimatroids
Redundancy and Helly

AbstractThe classical Helly’s Theorem about finite sets of convex sets is given an unusually simple proof based on a ‘Redundancy Lemma’. Because the proof is topological it extends immediately to a Helly’s Theorem for the well-known combinatorial topology representation of oriented matroids which is reviewed. The same proof is then used to strengthen Helly’s Theorem in a useful way relative to the Farkas Lemma, both for linear inequality systems and for topologically represented oriented matroids
Combinatorial and Algebraic Structure in Orlik–Solomon Algebras

AbstractThe Orlik–Solomon algebra A(G) of a matroid G is the free exterior algebra on the points, modulo the ideal generated by the circuit boundaries. On one hand, this algebra is a homotopy invariant of the complement of any complex hyperplane arrangement realizing G. On the other hand, some features of the matroid G are reflected in the algebraic structure of A(G). In this mostly expository article, we describe recent developments in the construction of algebraic invariants of A(G). We develop a categorical framework for the statement and proof of recently discovered isomorphism theorems which suggests a possible setting for classification theorems. Several specific open problems are formulated
Partitions of Points into Simplices withk-dimensional Intersection. Part I: The Conic Tverberg’s Theorem

AbstractTverberg’s 1966 theorem asserts that every set X of (m− 1)(d+ 1) + 1 points in Rdhas a partition X1, X2, . . . , Xmsuch that∩i=1mconvXi≠=φ. We give a short and elementary proof of a theorem on convex cones which generalizes this result. As a consequence, we deduce several divisibility properties, including the characterization of extremal sets which have no partition such that∩i=1mconvXiis at least one-dimensional and, in the particular cases m= 3 and m= 4, the proof of Reay’s conjecture that every set of (m− 1)(d+ 1) +k+ 1 points in general position in Rdhas a partition such that∩i=1mconvXiis at least k -dimensional