pinecone/core-2020-05-10-deduplication huggingface dataset:
On a Particular Class of Minihypers and Its Applications II. Improvements for q Square

AbstractA particular class of minihypers was studied previously by the authors (in press, Des. Codes Cryptogr.). For q square, this paper improves the results of that work, under the assumption that no weights occur in the minihyper. Using the link between these minihypers and maximal partial s-spreads of PG(N, q), (s+1) ∣(N+1), the findings on minihypers translate immediately into results on the extendability of partial s-spreads with small positive deficiency. Other applications of this characterisation of minihypers are given by P. Govaerts et al. (in press, European J. Combin.)
Multidimensional Ehrhart Reciprocity

AbstractIn an earlier paper (1999, Electron. J. Combin.6, R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved that, if our polytope is a simplex, the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. In the present paper we complete the picture by extending this result to general rational polytopes. As a corollary, we also generalize a reciprocity theorem of Stanley
On Quasi-thin Association Schemes

AbstractAn association scheme (or simply, a scheme) is called thin if each of its basic relations has valency 1. It is easy to see that thin schemes can be viewed as groups and, conversely, groups can be seen as thin schemes. In the present paper, we investigate schemes the basic relations of which have valency 1 or 2. We call these schemes quasi-thin. In order to formulate our results we let (X, R) denote a scheme (in the sense of P.-H. Zieschang). We first offer three sufficient conditions for (X, R) to have an automorphism group acting transitively on X. These conditions are (i) Oθ(R)∩Oθ(R)={1}, (ii) nOθ(R)=2, (iii) R possesses an element r such that 〈r〉=R and 〈rr*〉=〈r*r〉. We then prove that, if Oθ(R)=Oθ(R) and nOθ(R)=4, |X|/4∈{3, 4, 7, 8, 12, 16}. As a consequence of the latter result, we obtain a classification of the quasi-thin schemes with |X|=4p, where p is a prime number
Tree-like Properties of Cycle Factorizations

AbstractWe provide a bijection between the set of factorizations, that is, ordered (n−1)-tuples of transpositions in Sn whose product is (12…n), and labelled trees on n vertices. We prove a refinement of a theorem of J. Dénes (1959, Publ. Math. Inst. Hungar. Acad. Sci.4, 63–71) that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization corresponds naturally under our bijection to leaf edges (incident with a vertex of degree 1) of a tree. Moreover, we give a generalization of this fact
A Block Negacyclic Bush-Type Hadamard Matrix and Two Strongly Regular Graphs

AbstractA block negacyclic Bush-type Hadamard matrix of order 36 is used in a symmetric BGW(26, 25, 24) with zero diagonal over a cyclic group of order 12 to construct a twin strongly regular graph with parameters v=936, k=375, λ=μ=150 whose points can be partitioned in 26 cocliques of size 36. The same Hadamard matrix then is used in a symmetric BGW(50, 49, 48) with zero diagonal over a cyclic group of order 12 to construct a Siamese twin strongly regular graph with parameters v=1800, k=1029, λ=μ=588
Transitive Deficiency-One Baer Subgeometry Partitions

AbstractThe translation planes of order 43 with spread in PG(5, 4) arising from Baer subgeometry partitions of PG(2, 16) found by Mathon and Hamilton are characterized within all Baer subgeometry partitions admitting groups fixing one Baer subgeometry and acting transitively on the remaining Baer subgeometries. When q is even, we show that a Baer subgeometry partition of PG(2, q2) with a group fixing one and transitive on the remaining PG(2, q)'s forces the partition to be classical or q=4 and the partition is one of the partitions of Mathon and Hamilton. When q is odd, we give a complete characterization of these Baer subgeometry partitions. Further, generalizations are given for similar Baer subgeometry partitions of PG(2m, q2)
Forest Volume Decompositions and Abel–Cayley–Hurwitz Multinomial Expansions

AbstractThis paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R⊆S the polynomial enumerating out-degrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is∑r∈Rxr∑s∈Sxs|S|−|R|−1
Delta-Systems and Qualitative (In)Dependence

AbstractFollowing Erdős and Rado, three sets are said to form a delta triple if any two of them have the same intersection. Let F(n,3) denote the largest cardinality of a family of subsets of an n-set not containing a delta-triple. It is not known whether lim supn→∞n−1 log F(n,3)<1. We say that a family of bipartitions of an n-set is qualitatively 3/4-weakly 3-dependent if the common refinement of any 3 distinct partitions of the family has at least 6 non-empty classes (i.e., at least 3/4 of the total). Let I(n) denote the maximum cardinality of such a family. We derive a simple relation between the exponential asymptotics of F(n,3) and I(n) and show, as a consequence, that lim supn→∞n−1 log F(n,3)=1 if and only if lim sup n→∞n−1 log I(n)=1
Refined Stirling Numbers: Enumeration of Special Sets of Permutations and Set-Partitions

AbstractThe refined Stirling numbers of the first kindnm0m1 ··· mk−1 specify the number of permutations of n indices possessing mi cycles whose lengths modulo k are congruent to i, i=0,1,2,…,k−1. The refined Stirling numbers of the second kindnm0m1 ··· mk−1 are similarly defined in terms of set-partitions and the cardinalities of their disjoint blocks. Generating functions for these two types of refined Stirling numbers are derived using the Faà di Bruno formula. These generating functions allow the derivation of recurrence relations for both types of refined Stirling numbers
On Identifying Codes in Binary Hamming Spaces

AbstractA binary code C⊆{0,1}n is called r-identifying, if the sets Br(x)∩C, where Br(x) is the set of all vectors within the Hamming distance r from x, are all nonempty and no two are the same. Denote by Mr(n) the minimum possible cardinality of a binary r-identifying code in {0,1}n. We prove that if ρ∈[0,1) is a constant, then limn→∞n−1 log2M⌊ρn⌋(n)=1−H(ρ), where H(x)=−x log2x−(1−x) log2(1−x). We also prove that the problem whether or not a given binary linear code is r-identifying is Π2-complete
On k-wise set-intersections and k-wise Hamming-distances

AbstractWe prove a version of the Ray-Chaudhuri–Wilson and Frankl–Wilson theorems for k-wise intersections and also generalize a classical code-theoretic result of Delsarte for k-wise Hamming distances. A set of code-words a1, a2,…,ak of length n have k-wise Hamming-distance ℓ, if there are exactly ℓ such coordinates, where not all of their coordinates coincide (alternatively, exactly n−ℓ of their coordinates are the same). We show a Delsarte-like upper bound: codes with few k-wise Hamming-distances must contain few code-words
Reconstructing Infinite Sets of Integers

AbstractFor a set of integers A⊆Z and k⩾1 the k-deck of A is the function dA,k defined on sets S of k integers bydA,k(S)=∣{I∈Z∣ {s+I∣s∈S}⊆A}∣ . Our main result is that for k⩾3, a set for which the k-deck only takes finite values is determined up to translation by its k-deck and one finite non-zero value of its (k−1)-deck. This generalizes a result of Radcliffe and Scott (Electron. J. Combin.6 (1999), R20) which proved a weaker form of this statement for k=3. In order to establish this result, we generalize Kelly's Lemma for finite graphs to infinite sets of integers
A Polytopal Generalization of Sperner's Lemma

AbstractWe prove the following conjecture of Atanassov (Studia Sci. Math. Hungar.32 (1996), 71–74). Let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2,…,vn. Label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F. Then there are at least n−d full dimensional simplices of T, each labelled with d+1 different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in Alekseyevskaya (Discrete Math.157 (1996), 15–37) and Billera et al. (J. Combin. Theory Ser. B57 (1993), 258–268)
Partition Identities and a Theorem of Zagier

AbstractIn the study of partition theory and q-series, identities that relate series to infinite products are of great interest (such as the famous Rogers–Ramanujan identities). Using a recent result of Zagier, we obtain an infinite family of such identities that is indexed by the positive integers. For example, if m=1, then we obtain the classical Eisenstein series identity ∑λ⩾1 odd(−1)(λ−1)/2qλ(1−q2λ)=q∏n=1∞(1−q8n)4(1−q4n)2 . If m=2 and (·3) denotes the usual Legendre symbol modulo 3, then we obtain ∑λ⩾1(λ3)qλ(1−q2λ)=q∏n=1∞(1−qn)(1−q6n)6(1−q2n)2(1−q3n)3 . We describe some of the partition theoretic consequences of these identities. In particular, we find simple formulas that solve the well-known problem of counting the number of representations of an integer as a sum of an arbitrary number of triangular numbers
Some Observations on Dyson's New Symmetries of Partitions

AbstractWe utilize Dyson' concept of the adjoint of a partition to derive an infinite family of new polynomial analogs of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank ⩽k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's “modular” partitions with modulus 2. This way we find a new combinatorial proof of Gauss's famous identity. We give a direct combinatorial proof that for n>1 the partitions of n with crank k are equinumerous with partitions of n with crank −k
Circulant 16-Modular Hadamard Matrices and Jacobi Sums

AbstractWe are concerned here with the existence problem of 16-modular circulant Hadamard matrices H of size 4p (p prime), satisfying the additional condition that any two rows at distance n/2 in H are strictly orthogonal. A necessary existence condition is p≡1 mod 8. For p≡1 mod 16, existence follows from the more general result of a previous paper of ours, showing the existence of (p−1)-modular matrices of the above kind. In the remaining case p≡9 mod 16, we construct explicit examples which solve the problem whenever 2 is a fourth power mod p. When 2 is not a fourth power mod p, we conjecture that such matrices cannot exist
On the Turán Number of Triple Systems

AbstractFor a family of r-graphs F the Turán number ex (n, F) is the maximum number of edges in an n vertex r-graph that does not contain any member of F. The Turán density π(F)=limn→∞ex(n,F)((nr). When F is an r-graph, π(F)≠0, and r>2, determining π(F) is a notoriously hard problem, even for very simple r-graphs F. For example, when r=3, the value of π(F) is known for very few (<10) irreducible r-graphs. Building upon a method developed recently by de Caen and Füredi (J. Combin. Theory Ser. B78 (2000), 274–276), we determine the Turán densities of several 3-graphs that were not previously known. Using this method, we also give a new proof of a result of Frankl and Füredi (Combinatorica 3 (1983), 341–349) that π(H)=29, where H has edges 123,124,345. Let F(3,2) be the 3-graph 123,145,245,345, let K−4 be the 3-graph 123,124,234, and let C5 be the 3-graph 123,234,345,451,512. We prove &#x02022;⩽(F(3,2))⩽12,&#x02022;({:K,C})⩽=0.322581,&#x02022;0.464<(C)⩽2−√2<0.586.
 The middle result is related to a conjecture of Frankl and Füredi (Discrete Math.50 (1984) 323–328) that π(K−4)=27. The best known bounds are 27⩽π(K−4)⩽13
On the Cardinality of Intersection Sets in Inversive Planes

AbstractIntersection sets and blocking sets play an important role in contemporary finite geometry. There are cryptographic applications depending on their construction and combinatorial properties. This paper contributes to this topic by answering the question: how many circles of an inversive plane will be blocked by a d-element set of points that has successively been constructed using a greedy type algorithm? We derive a lower bound for this number and thus obtain an upper bound for the cardinality of an intersection set of smallest size. Defining a coefficient called greedy index, we finally give an asymptotic analysis for the blocking capabilities of circles and subplanes of inversive planes
Enumeration of Lozenge Tilings of Hexagons with Cut-Off Corners

AbstractMotivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases” removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case
On Two-Intersection Sets with Respect to Hyperplanes in Projective Spaces

AbstractIn [Blokhuis and Lavrauw (Geom. Dedicata81 (2000), 231–243)] a construction of a class of two-intersection sets with respect to hyperplanes in PG(r−1,qt), rt even, is given, with the same parameters as the union of (qt/2−1)/(q−1) disjoint Baer subgeometries if t is even and the union of (qt−1)/(q−1) elements of an (r/2−1)-spread in PG(r−1,qt) if t is odd. In this paper, we prove that although they have the same parameters, they are different. This was previously proved in [Ball et al. (Finite Fields Appl.6 (2000), 294–301)] in the special case where r=3 and t=4
Asymmetric Binary Covering Codes

AbstractAn asymmetric binary covering code of length n and radius R is a subset C of the n-cube Qn such that every vector x∈Qn can be obtained from some vector c∈C by changing at most R 1's of c to 0's, where R is as small as possible. K+(n,R) is defined as the smallest size of such a code. We show K+(n,R)∈Θ(2n/nR) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K+(n,n−R̄)=R+1 for constant coradius R iff n⩾R(R+1)/2. These two results are extended to near-constant R and R, respectively. Various bounds on K+ are given in terms of the total number of 0's or 1's in a minimal code. The dimension of a minimal asymmetric linear binary code ([n,R]+-code) is determined to be min{0,n−R}. We conclude by discussing open problems and techniques to compute explicit values for K+, giving a table of best-known bounds
On the Natural Representation of S(Ω) into L2(P(Ω)): Discrete Harmonics and Fourier Transform

AbstractLet Ω denote a nonempty finite set. Let S(Ω) denote the symmetric group on Ω and let P (Ω) denote the power set of Ω. Let ρ : S(Ω)→U(L2(P (Ω))) be the left unitary representation of S(Ω) associated with its natural action on P (Ω). We consider the algebra consisting of those endomorphisms of L2(P (Ω)) which commute with the action of ρ. We find an attractive basis B for this algebra. We obtain an expression, as a linear combination of B, for the product of any two elements of B. We obtain an expression, as a linear combination of B, for the adjoint of each element of B. It turns out that the Fourier transform on P(Ω) is an element of our algebra; we give the matrix which represents this transform with respect to B
Degree 8 Maximal Arcs in PG(2,2h), h Odd

AbstractIn a recent paper R. Mathon gave a new construction method for maximal arcs in finite Desarguesian projective planes that generalised a construction of Denniston. He also gave several instances of the method to construct new maximal arcs. In this paper, the structure of the maximal arcs is examined to give geometric and algebraic methods for proving when the maximal arcs are not of Denniston type. New degree 8 maximal arcs are also constructed in PG(2,2h), h⩾5, h odd. This, combined with previous results, shows that every Desarguesian projective plane of (even) order greater that 8 contains a degree 8 maximal arc that is not of Denniston type
Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic Models

AbstractThis article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gröbner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further, we show how to build Gröbner bases of a reducible model from the Gröbner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Gröbner bases. Finally, we present the solution of a problem posed by Vlach (Discrete Appl. Math.13 (1986) 61–78) concerning the dimension of fibers coming from models corresponding to the boundary of a simplex
Permutations with Low Discrepancy Consecutive k-sums

AbstractConsider the permutation π=(π1,…, πn) of 1,2,…, n as being placed on a circle with indices taken modulo n. For given k⩽n there are n sums of k consecutive entries. We say the maximum difference of any consecutive k-sum from the average k-sum is the discrepancy of the permutation. We seek a permutation of minimum discrepancy. We find that in general the discrepancy is small, never more than k+6, independent of n. For g= gcd(n,k)>1, we show that the discrepancy is ⩽72. For g=1 it is more complicated. Our constructions show that the discrepancy never exceeds k/2 by more than 9 for large n, while it is at least k/2 for infinitely many n.We also give an analysis for the easier case of linear permutations, where we view the permutation as written on a line. The analogous discrepancy is at most 2 for all n,k
The Rank and Minimal Border Strip Decompositions of a Skew Partition

AbstractThe rank of an ordinary partition of a nonnegative integer n is the length of the main diagonal of its Ferrers or Young diagram. Nazarov and Tarasov gave a generalization of this definition for skew partitions and proved some basic properties. We show the close connection between the rank of a skew partition λ/μ and the minimal number of border strips whose union is λ/μ. A general theory of minimal border strip decompositions is developed and an application is given to the evaluation of certain values of irreducible characters of the symmetric group
Polyhedra of Small Order and Their Hamiltonian Properties

AbstractWe describe the results of an enumeration of several classes of polyhedra. The enumerated classes include polyhedra with up 13 vertices, simplicial polyhedra with up to 16 vertices, 4-connected polyhedra with up to 15 vertices, non-Hamiltonian polyhedra with up to 15 vertices, bipartite polyhedra with up to 24 vertices, and bipartite trivalent polyhedra with up to 44 vertices. The results of the enumeration were used to systematically search for certain smallest non-Hamiltonian polyhedral graphs. In particular, the smallest non-Hamiltonian planar graphs satisfying certain toughness-like properties are presented here, as are the smallest non-Hamiltonian, 3-connected, Delaunay tessellations and triangulations. Improved upper and lower bounds on the size of the smallest non-Hamiltonian, inscribable polyhedra are also given
Dichromatic Sums Revisited

AbstractIn a previous paper the author studied an enumerating power seriesΦin six variables. The typical term was the sum of a dichromatic polynomial over all rooted planar maps of given numbers of vertices and faces and given valencies for the root face and root vertex. Loops and multiple joins were allowed. The polynomial was the one now commonly called the “Tutte polynomial” by other writers. An equation forΦwas obtained. It made possible a recursive calculation of coefficients in the order of increasing edge number. The present paper arose out of the observation that the equation forΦtakes a particularly simple form when the variablesxandyof the dichromatic polynomial are each given the value 1. The value of the polynomial is then the number of spanning trees of the map concerned. In that special case a theoretical solution is obtained. It is stated in terms of a remainder obtained when a certain power series in the four remaining variables, slightly transformed, is divided by a certain polynomial
Orthogonal A-Trails of 4-Regular Graphs Embedded in Surfaces of Low Genus

AbstractAnton Kotzig has shown that every connected 4-regular plane graph has an A-trail, that is an Euler trail in which any two consecutive edges lie on a common face boundary. We shall characterise the 4-regular plane graphs which contain twoorthogonalA-trails, that is to say two A-trails for which no subtrail of length 2 appears in both A-trails. Our proof gives rise to a polynomial algorithm for deciding if two such A-trails exists. We shall also discuss the corresponding problem for graphs in the projective plane and the torus, and the related problem of deciding when a 2-regular digraph contains two orthogonal Euler trails
RecognizingP3-Structure: A Switching Approach

AbstractAP3is a set of three vertices of a graph that induces a chordless path; theP3-structure of a graph is the set of allP3's. Chvátal asked if there is a polynomial time algorithm to determine whether an arbitrary three-uniform hypergraph is theP3-structure of some graph. The answer is yes: we describe such an algorithm. Our algorithm is based on switching (as defined by Seidel), a characterization of graphs with the sameP3-structure, a sufficient condition forP3-structure uniqueness, and a reduction of a certain partitioning problem to two-satisfiability
Entropy Splitting Hypergraphs

AbstractHypergraph entropy is an information theoretic functional on a hypergraph with a probability distribution on its vertex set. It is sub-additive with respect to the union of hypergraphs. In case of simple graphs, exact additivity for the entropy of a graph and its complement with respect to every probability distribution on the vertex set gives a characterization of perfect graphs. Here we investigate uniform hypergraphs with an analoguous behaviour of their entropy. The main result is the characterization of 3-uniform hypergraphs having this entropy splitting property. It is also shown that fork⩾4 no non-trivialk-uniform hypergraph has this property
Vertex-Switching Reconstruction and Folded Cubes

AbstractIn this note we use eigenvalues of folded cubes to simplify an analogue of Kelly's Lemma for vertex-switching reconstruction due to Krasikov and Roditty. Our new version states that the number of subgraphs (or induced subgraphs) of ann-vertex graphGisomorphic to a givenm-vertex graph can be found from thes-vertex-switching deck ofGprovided the Krawtchouk polynomialKsnhas no even roots in [0, m]. This generalizes a condition of Stanley fors-vertex-switching reconstructibility. We also comment on the role of cubes and folded cubes in the theory of vertex-switching reconstruction