pinecone/core-2020-05-10-deduplication huggingface dataset:
Upper Bounds on Permutation Codes via Linear Programming

AbstractAn upper bound on permutation codes of lengthnis given. This bound is a solution of a certain linear programming problem and is based on the well-developed theory of association schemes. Several examples are presented. For instance, the 255 values of the bound forn≤ 8 are tabulated. It turns out that, forn≤ 8, the Kiyota bound for group codes also holds for unrestricted codes at least in 178 cases. Also an easier (but weaker) polynomial version of the bound is given. It is obtained by showing that the mappingsFk(θ) (0≤k≤n/2), whereFkis the Charlier polynomial of degreekand θ the natural character of the symmetric groupSn, are mutually orthogonal characters ofSn
A Kruskal–Katona Type Theorem for the Linear Lattice

AbstractWe present an analog of the well-known Kruskal–Katona theorem for the poset of subspaces of PG(n,2) ordered by inclusion. For givenk,ℓ (k<ℓ) andmthe problem is to find a family of sizemin the set of ℓ-subspaces of PG(n,2), containing the minimal number ofk-subspaces. We introduce two lexicographic type orders O1and O2on the set of ℓ-subspaces, and prove that the firstmof them, taken in the order O1, provide a solution in the casek=0 and arbitrary ℓ>0, and one taken in the order O2, provide a solution in the case ℓ=n−1 and arbitraryk<n−1. Concerning other values ofkand ℓ, we show that forn≥ 3 the considered poset is not Macaulay by constructing a counterexample in the case ℓ=2 andk=1
An Interpretation for the Tutte Polynomial

AbstractFor any matroidMrealizable over Q, we give a combinatorial interpretation of the Tutte polynomialTM(x,y) which generalizes many of its known interpretations and specializations, including Tutte's coloring and flow interpretations ofTM(1−t,0),TM(0,−t); Crapo and Rota's finite field interpretation ofTM(1−qk,0); the interpretation in terms of the Whitneycorank-nullitypolynomial; Greene's interpretation as the weight enumerator of a linear code and its recent generalization to higher weight enumerators by Barg; Jaeger's interpretation in terms of linear code words and dual code words with disjoint support; and Brylawksi and Oxley's two-variable coloring formula
Gaussian, Strong and Transversal Greedoids

AbstractVarious types of greedoids have been studied in relation to the Greedy Algorithm and therefore it is not surprising that these structures are inter-linked in various ways. However, it is quite surprising that the Gauss greedoids and the strong greedoids, approached from quite different points of view, turn out to be precisely the same. Although this can be seen very indirectly by combining the two sets of separate results, the main purpose of this short paper is to prove directly that the two structures are identical, thus giving a simple axiomatic characterisation of the Gauss greedoids. We also note that the transversal greedoids provide an interesting class of examples of such greedoids
On the Isomorphism Problem for Finite Cayley Graphs of Bounded Valency

AbstractFor a subsetSof a groupGsuch that 1∉SandS=S−1, the associated Cayley graph Cay(G,S) is the graph with vertex setGsuch that {x,y} is an edge if and only ifyx−1∈S. Each σ∈Aut(G) induces an isomorphism from Cay(G,S) to the Cayley graph Cay(G,Sσ). For a positive integerm, the groupGis called anm-CI-group if, for all Cayley subsetsSof size at mostm, whenever Cay(G,S) ≅Cay(G,T) there is an element σ∈Aut(G) such thatSσ=T. It is shown that ifGis anm-CI-group for somem≥4, thenG=U×V, where (|U|,|V|) =1,Uis abelian, andVbelongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification
Regular Honest Graphs, Isoperimetric Numbers, and Bisection of Weighted Graphs

AbstractThe edge-integrity of a graph G isI′ (G) :=min{| S | +m(G−S) : S⊂E }, where m(H) denotes the maximum order of a component of H. A graph G is calledhonest if its edge-integrity is the maximum possible; that is, equals the order of the graph. The only honest 2-regular graphs are the 3-, 4-, and 5-cycles. Lipman13proved that there are exactly twenty honest cubic graphs. In this paper we exploit a technique of Bollobás8,9to prove that for every k≥ 6, almost all k -regular graphs are honest. On the other hand, we show that there are only finitely many 4-regular honest graphs. To prove this, we use a weighted version of the upper bound on the isoperimetric number due to Alon1. We believe that this version is of interest by itself
Characterizing Combinatorial Geometries by Numerical Invariants

AbstractWe show that the projective geometry PG(r− 1,q ) for r& 3 is the only rank- r(combinatorial) geometry with (qr− 1)/(q− 1) points in which all lines have at least q+ 1 points. For r= 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice
Shortest Paths in Distance-regular Graphs

AbstractWe aim here to introduce a new point of view of the Laplacian of a graph, Γ. With this purpose in mind, we consider L as a kernel on the finite space V(Γ), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties such as maximum and energy principles and the equilibrium principle on any proper subset of V(Γ). If Γ is a proper set of a suitable host graph, then the equilibrium problem for Γ can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of Γ. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs
Affine Distance-transitive Groups of Dimension One

AbstractLet V be a one-dimensionalFq-space and G be a subgroup in AΓL(V) ∼ =AΓL(1, q) containing the translations and acting primitively on the set of vectors in V. Let G0be the stabilizer in G of the zero vector, so thatG0≤ΓL(V) ∼ =ΓL(1, q) and G=V:G0 . Suppose that Γ is a graph structure on V on which G acts distance-transitively. Such graphs of diameter at most 2 are known from the literature. Here, we show that if Γ has diameter at least 3, then q= 64, G0 ∼ =Z9: Z3or Z9: Z6and Γ is the Hamming graph H(4, 3)
A Monoid for the Universal K-Bruhat Order

Structure constants for the multiplication of Schubert polynomials by Schur
symmetric polynomials are known to be related to the enumeration of chains in a
new partial order on S_\infty, which we call the universal k-Bruhat order. Here
we present a monoid M for this order and show that $M$ is analogous to the
nil-Coxeter monoid for the weak order on S_\infty. For this, we develop a
theory of reduced sequences for M. We use these sequences to give a
combinatorial description of the structure constants above. We also give
combinatorial proofs of some of the symmetry relations satisfied by these
structure constants.Comment: LaTeX-2e, 21 pages, 3 figures, uses epsf.st
Circular Chromatic Numbers of Distance Graphs with Distance Sets Missing Multiples

AbstractGiven positive integers m, k, s withm&sk, let Dm, k, srepresent the set {1, 2,⋯ , m } \ { k, 2 k,⋯ , sk }. The distance graph G(Z, Dm, k, s) has as vertex set all integersZ and edges connecting i and j whenever | i−j | ∈Dm, k, s. This paper investigates chromatic numbers and circular chromatic numbers of the distance graphs G(Z, Dm, k,s). Deuber and Zhu and Liu have shown that⌈m+sk+ 1s+ 1⌉ ≤χ(G(Z, Dm, k,s)) ≤ ⌈m+sk+ 1s+ 1⌉ +1whenm≥ (s+ 1)k. In this paper, by establishing bounds for the circular chromatic number χc(G(Z, Dm,k , s)) of G(Z, Dm, k,s), we determine the values of χ(G(Z, Dm,k , s)) for all positive integers m, k, s andχc (G(Z, Dm, k,s)) for some positive integers m, k, s
A Class of Lattices Whose Intervals are Spherical or Contractible

AbstractWe study a class of lattices called weak*complemented lattices which are shown to have the property that the order complex of any interval of the lattice is either contractible or homotopy equivalent to a sphere. The two main examples are lattices generated by intervals in a total order and the lattices of partitions of integers under dominance order. The proofs are done mainly using homotopy complementation formulas for lattices and with a method called B-labeling. We also show that a class of lattices called Greene lattices are either contractible or spherical. Lattices generated by multisets are also discussed
Cayley Digraphs from Complete Generalized Cycles

AbstractThe complete generalized cycleG(d,n) is the digraph which hasZn×Zdas the vertex set and every vertex (i,x) is adjacent to thedvertices (i+ 1,y) withy∈Zd. As a main result, we give a necessary and sufficient condition for the iterated line digraphG(d,n,k) =Lk−1G(d,n), withda prime number, to be a Cayley digraph in terms of the existence of a groupΓdof orderdand a subgroupNof (Γd)nisomorphic to (Γd)k. The condition is shown to be also sufficient for any integerd≥ 2. IfΓdis a ringRandNis a submodule ofRn, it is said thatG(d,n,k) is anR-Cayley digraph. By using some properties of the homogeneous linear recurrences in finite rings, necessary and sufficient conditions forG(d,n,k) to be anR-Cayley digraph are obtained. As a consequence, whenR=Zda new characterization for the digraphsG(d,n,k) to beZd-Cayley digraphs is derived. As a corollary, sufficient conditions for the corresponding underlying graphs to be Cayley can be deduced. Ifdis a prime power andFdis a finite field of orderd, the digraphsG(d,n,k) which areFd-Cayley digraphs are in 1-1 correspondence with the cyclic (n,k)-linear codes overFd
Zonotopes, Dicings, and Voronoi’s Conjecture on Parallelohedra

AbstractIn 1909, Voronoi conjectured that if some selection of translates of a polytope forms a facet-to-facet tiling of euclidean space, then the polytope is affinely equivalent to the Voronoi polytope for a lattice. He referred to polytopes with this tiling property as parallelohedra, but they are now frequently called parallelotopes. I show that Voronoi’s conjecture holds for the special case where the parallelotope is a zonotope. I also show that the Voronoi polytope for a lattice is a zonotope if and only if the Delaunay tiling for the lattice is a dicing (defined at the beginning of Section 3)
On Commutation Classes of Reduced Words in Weyl Groups

AbstractThis paper is concerned with commutation classes of reduced words in Weyl groups. It is divided in three sections. In the first, we give a recursive formula for the number of reduced words in a commutation class. In the second, we give a tableau-like description of all the reduced words adapted to a quiver in the case of simply laced root system. In the last, we consider the case of the longest element w0for the symmetric group S5and illustrate the fact that the set of commutation classes of reduced words of w0have nice symmetries and a topological structure
Maximal Unimodular Systems of Vectors

AbstractA subset R of a vector space V(orRn) is called unimodular(or U -system) if every vector r∈R has an integral representation in every basis B⊆R. A U -system R is calledmaximal if one cannot add a non-zero vector not colinear to vectors of R such that the new system is unimodular and spans RR. In this work, we refine assertions of Seymour7and give a description of maximal U -systems. We show that a maximal U -system can be obtained as amalgams (as 1- and 2-sums) of simplest maximal U -systems called components. A component is a maximal U -system having no 1- and 2-decompositions. It is shown that there are three types of components: the root systems An, which are graphic, cographic systems related to non-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a special system E5representing the matroid R10from7which is neither graphic nor cographic. We give conditions that are necessary and sufficient for maximality of an amalgamated U -system. We give a complete description of all 11 maximal U -systems of dimension 6
Geometrical Techniques for Estimating Numbers of Linear Extensions

AbstractLetPbe a two-dimensional order, and __Pany complement ofP, i.e., any partial order whose comparability graph is the complement of the comparability graph ofP. Lete(Q) denote the number of linear extensions of the partial orderQ. Sidorenko showed thate(P)e(__P) ≥n!, for any two-dimensional partial orderP. In this note, we use results from polyhedral combinatorics, and from the geometry ofRn, to give a companion upper bound one(P)e(__P), as well as an alternative proof of the lower bound. We use these results to obtain bounds on the number of linear extensions of a random two-dimensional partial order
The Geodetic Number of an Oriented Graph

AbstractFor two vertices u and v of an oriented graph D, the set I(u, v) consists of all vertices lying on a u−v geodesic or v−u geodesic in D. If S is a set of vertices of D, then I(S) is the union of all sets I(u, v) for vertices u and v in S. The geodetic number g(D) is the minimum cardinality among the subsets S of V(D) with I(S) =V(D). Several results concerning the geodetic numbers of connected oriented graphs are presented. For a nontrivial connected graph G, the lower orientable geodetic number g−(G) of G is the minimum geodetic number among the orientations of G and the upper orientable geodetic number g+(G) is the maximum such geodetic number. It is shown that g−(G)≠g+(G) for every connected graph of order at least 3. The lower and upper orientable geodetic numbers of several well known classes of graphs are determined. It is shown that for every two integers n and m with 1 ≤n− 1 ≤m≤ (n 2) , there exists a connected graph G of order n and size m such that g+(G) =n
Weakly Flag-transitive Configurations and Half-arc-transitive Graphs

AbstractA configuration is weakly flag-transitive if its group of automorphisms acts intransitively on flags but the group of all automorphisms and anti-automorphisms acts transitively on flags. It is shown that weakly flag-transitive configurations are in one-to-one correspondence with bipartite12-arc-transitive graphs of girth not less than 6. Several infinite families of weakly flag-transitive configurations are given via their Levi graphs. Among others an infinite family of non-self-polar weakly flag-transitive configurations is constructed. The smallest known weakly flag-transitive configuration has 27 points and the smallest known non-self-polar weakly flag-transitive configuration has 34 points
Speed’s Orbit Problem: Variations on Themes of Burnside

AbstractThe paper offers a solution to T. P. Speed’s Orbit Problem, describing the orbit decomposition of direct powers of a transitive finite group action. The solution is given in terms of the Burnside algebra of the group and the right action of an incidence algebra of the subgroup lattice on a module of finitary functions. As a corollary, it is shown that almost all orbits in powers of a faithful action are regular. This gives one possible ‘GF(1)’ version of Burnside’s original theorem that each irreducible complex representation of a finite group appears in a tensor power of a fixed faithful representation
The Local Structure of a Bipartite Distance-regular Graph

AbstractIn this paper, we consider a bipartite distance-regular graph Γ= (X, E) with diameter d≥ 3. We investigate the local structure ofΓ , focusing on those vertices with distance at most 2 from a given vertex x. To do this, we consider a subalgebra R=R(x) ofMat0307a0x.gif X(C), where 0307a1x.gifX denotes the set of vertices in X at distance 2 from x. R is generated by matrices Ã, 0307a2x.gif J, and 0307a3x.gif D defined as follows. For all y, z∈ 0307a4x.gif X, the (y,z )-entry of Ã is 1 if y, z are at distance 2, and 0 otherwise. The (y, z)-entry of 0307a5x.gif J equals 1, and the (y,z )-entry of 0307a6x.gif D equals the number of vertices of X adjacent to each ofx , y, and z. We show that R is commutative and semisimple, with dimension at least 2. We assume thatdimR is one of 2, 3, or 4, and explore the combinatorial implications of this. We are motivated by the fact that if Γ has a Q-polynomial structure, thendimR≤ 4
On the Ring of Simultaneous Invariants for the Gleason–MacWilliams Group

AbstractWe construct a canonical generating set for the polynomial invariants of the simultaneous diagonal action (of arbitrary number of l factors) of the two-dimensional finite unitary reflection group G of order 192, which is called the group No. 9 in the list of Shephard and Todd, and is also called the Gleason–MacWilliams group. We find this canonical set in the vector space (⊗i=1lV)G, where V denotes the (dual of the) two-dimensional vector space on which the group G acts, by applying the techniques of Weyl (i.e., the polarization process of invariant theory) to the invariants C [ x, y ]G0of the two-dimensional group G0of order 48 which is the intersection of G and SL(2, C). It is shown that each element in this canonical set corresponds to an irreducible representation which appears in the decomposition of the action of the symmetric group Sl. That is, by letting the symmetric group Slacts on each element of the canonical generating set, we get an irreducible subspace on which the symmetric group Slacts irreducibly, and all these irreducible subspaces give the decomposition of the whole space (⊗i=1lV)G. This also makes it possible to find the generating set of the simultaneous diagonal action (of arbitrary l factors) of the group G. This canonical generating set is different from the homogeneous system of parameters of the simultaneous diagonal action of the group G. We can construct Jacobi forms (in the sense of Eichler and Zagier) in various ways from the invariants of the simultaneous diagonal action of the group G, and our canonical generating set is very fit and convenient for the purpose of the construction of Jacobi forms
Multidiameters and Multiplicities

AbstractThe k -diameter of a graph Γ is the largest pairwise minimum distance of a set ofk vertices in Γ, i.e., the best possible distance of a code of size k in Γ. Ak -diameter for some k is called a multidiameter of the graph. We study the function N(k,Δ , D), the largest size of a graph of degree at most Δ and k -diameter D. The graphical analogues of the Gilbert bound and the Hamming bound in coding theory are derived. Constructions of large graphs with given degree and k -diameter are given. Eigenvalue upper bounds are obtained. By combining sphere packing arguments and eigenvalue bounds, new lower bounds on spectral multiplicity are derived. A bound on the error coefficient of a binary code is given
Completely Transitive Codes in Hamming Graphs

AbstractA code in a graph Γ is a non-empty subset C of the vertex set V of Γ. Given C, the partition of V according to the distance of the vertices away from C is called the distance partition of C. A completely regular code is a code whose distance partition has a certain regularity property. A special class of completely regular codes are the completely transitive codes. These are completely regular codes such that the cells of the distance partition are orbits of some group of automorphisms of the graph. This paper looks at these codes in the Hamming Graphs and provides a structure theorem which shows that completely transitive codes are made up of either transitive or nearly complete, completely transitive codes. The results of this paper suggest that particular attention should be paid to those completely transitive codes of transitive type
Algebras Related to Matroids Represented in Characteristic Zero

AbstractLet k be a field of characteristic zero. We consider graded subalgebras A ofk [ x1,⋯ , xm]/ (x12,⋯ , xm2) generated by d linearly independent linear forms. Representations of matroids over k provide a natural description of the structure of these algebras. In return, the numerical properties of the Hilbert function of A yield some information about the Tutte polynomial of the corresponding matroid. Isomorphism classes of these algebras correspond to equivalence classes of hyperplane arrangements under the action of the general linear group
Cyclic Polytopes and Oriented Matroids

AbstractConsider the moment curve in the real euclidean space Rddefined parametrically by the map γ: R→Rd,t∣→γ(t) = (t, t2,⋯ , td). The cyclic d -polytopeCd (t1,⋯ , tn) is the convex hull ofn&d different points on this curve. The matroidal analogs are the alternating oriented uniform matroids. A polytope (resp. matroid polytope) is called cyclic if its face lattice is isomorphic to that ofCd (t1,⋯ , tn). We give combinatorial and geometrical characterizations of cyclic (matroid) polytopes. A simple evenness criterion determining the facets ofCd (t1,⋯ , tn) was given by Gale . We characterize the admissible orderings of the vertices of the cyclic polytope, i.e., those linear orderings of the vertices for which Gale’s evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes
The Generalized Baues Problem for Cyclic Polytopes I

AbstractAn important special case of the generalized Baues problem asks whether the order complex of all proper polyhedral subdivisions of a given point configuration, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first non-trivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertices for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many non-regular triangulations of cyclic polytopes. Beyond this, we find triangulations of C(11, 5) with flip deficiency. This proves—among other things—that there are triangulations of cyclic polytopes that are non-regular for every choice of points on the moment curve
Simple 0/1-Polytopes

AbstractFor general polytopes, it has turned out that with respect to many questions it suffices to consider only the simple polytopes, i.e., d -dimensional polytopes where every vertex is contained in only d facets. In this paper, we show that the situation is very different within the class of 0/1-polytopes, since every simple 0/1-polytope is the (cartesian) product of some 0/1-simplices (which proves a conjecture of Ziegler), and thus, the restriction to simple 0/1-polytopes leaves only a very small class of objects with a rather trivial structure
On the Number of Edges in Hypergraphs Critical with Respect to Strong Colourings

AbstractA colouring of the vertices of a hypergraph G is called strong if, for every edgeA , the colours of all vertices in A are distinct. It corresponds to a colouring of the generated graphΓ(G) obtained from G by replacing every edge by a clique. We estimate the minimum number of edges possible in a k -critical t -uniform hypergraph with a given number of vertices. In particular we show that, for k≥t+ 2, the problem reduces in a way to the corresponding problem for graphs. In the case when the generated graph of the hypergraph has bounded clique number, we give a lower bound that is valid for sufficiently large k and is asymptotically tight in k; this bound also holds for list strong colourings