pinecone/core-2020-05-10-deduplication huggingface dataset:
Explicit Constructions of Perfect Hash Families from Algebraic Curves over Finite Fields

AbstractLet A be a set of order n and B be a set of order m. An (n, m, w)-perfect hash family is a set H of functions from A to B such that for any X⊆A with |X|=w, there exists an element h∈H such that h is one-to-one when restricted to X. Perfect hash families have many applications to computer science, such as database management, circuit complexity theory and cryptography. In this paper, we provide explicit constructions of perfect hash families based on algebraic curves over finite fields. In particular, using the Garcia–Stichtenoth curves, we obtain infinite classes of (n, m, w)-perfect hash families with |H|=O(logn) for fixed m and w, which are among the most efficient explicit constructions for perfect hash families known in the literature. We also exhibit examples to show the efficiency of the new constructions and their applications to the constructions of cover-free families
Root Sets of Polynomials Modulo Prime Powers

AbstractA subset R of the integers modulo n is defined to be a root set if it is the set of roots of some polynomial. Using the Chinese Remainder Theorem, the question of finding and counting root sets modn is reduced to finding root sets modulo a prime power. In this paper, we provide a recursive construction for root sets modulo a prime power. We use this recursion to show that the number of root sets modulo pk for fixed k is a polynomial in p, raised to the pth power. Moreover, we show that the leading term of this polynomial is ckp⌊k2/4⌋ where ck=(k2!)−1 if k is even and ck=(k−12!)−1+(k+12!)−1 if k is odd, thus giving an asymptotic estimate on the number of root sets for fixed k. Finally, we generalize these results to arbitrary Dedekind domains
A Characterization of (3+1)-Free Posets

AbstractPosets containing no subposet isomorphic to the disjoint sums of chains 3+1 and/or 2+2 are known to have many special properties. However, while posets free of 2+2 and posets free of both 2+2 and 3+1 may be characterized as interval orders, no such characterization is known for posets free of only 3+1. We give here a characterization of (3+1)-free posets in terms of their antiadjacency matrices. Using results about totally positive matrices, we show that this characterization leads to a simple proof that the chain polynomial of a (3+1)-free poset has only real zeros
Completed GPS Covers All Bent Functions

AbstractIn a recent paper, the so-called “generalized partial spread” (GPS) class which unifies almost all the known classes of bent functions is introduced. A necessary condition for a bent function to belong to GPS is that it takes the same value as its dual at the zero vector. In this paper, it is shown that the necessary condition above is sufficient. This proves that the completed class by composition with translations covers all the binary bent functions. Moreover, the elements of GPS are characterized in term of solutions of a quadratic Diophantine equation which may lead to count all bent functions. These results are presented in the general framework of partial bent functions which unify bent functions and r-dimensional vector space indicators
Carathéodory's Theorem and H-Convexity

AbstractIn 1976, V. Boltyanski introduced the functional md for compact, convex bodies. With the help of this functional, some theorems of combinatorial geometry were derived. For example, the first author obtained a Helly-type theorem, later some particular cases of the Szökefalvi–Nagy problem were resolved. Further on, exact estimates for the cardinalities of primitive fixing and hindering systems of compact, convex bodies were established, etc. In this article, we discuss the connection of the classical Carathéodory Theorem with the functional md
Efficient Reconstruction of Sequences from Their Subsequences or Supersequences

AbstractIn the paper two combinatorial problems for the set Fnq of sequences of length n over the alphabet Fq={0, 1, …, q−1} are considered. The maximum size N−q(n, t) of the set of common subsequences of length n−t and the maximum size N+q(n, t) of the set of common supersequences of length n+t of two different sequences of Fnq are found for any nonnegative integers n and t. The number N−q(n, t)+1 (respectively, N+q(n, t)+1) is equal to the minimum number N of different subsequences of length n−t (supersequences of length n+t) of an unknown sequence X∈Fnq which are sufficient for its reconstruction. Simple algorithms to recover X∈Fnq from N−q(n, t)+1 of its subsequences of length n−t and from N+q(n, t)+1 of its supersequences of length n+t are given
Generalized Binomial Coefficients for Molecular Species

AbstractLet ξ be a complex variable. We associate a polynomial in ξ, denoted (MN)ξ, to any two molecular species M=M(X) and N=N(X) by means of a binomial-type expansion of the formM(ξ+X)=∑NMNξN(X).In the special case M(X)=Xm, the species of linear orders of length m, the above formula reduces to the classical binomial expansion(ξ+X)m=∑nmnξm−nXn.When ξ=1, a M(1+X)-structure can be interpreted as a partially labelled M-structure and (MN)1 is a nonnegative integer, denoted (MN) for simplicity. We develop some basic properties of these “generalized binomial coefficients” and apply them to study solutions, Φ, of combinatorial equations of the form M(Φ)=Ψ in the context of C-species, M being molecular and Ψ being a given C-species. This generalizes the study of symmetric square roots (where M=E2, the species of 2-element sets) initiated by P. Bouchard, Y. Chiricota, and G. Labelle in (1995, Discrete Math.139, 49–56)
Multiple Left Regular Representations Generated by Alternants

AbstractLet p1>…>pn⩾0, and Δp=det‖xpji‖ni, j=1. Let Mp be the linear span of the partial derivatives of Δp. Then Mp is a graded Sn-module. We prove that it is the direct sum of graded left regular representations of Sn. Specifically, set λj=pj−(n−j), and let Ξλ(t) be the Hilbert polynomial of the span of all skew Schur functions sλ/μ as μ varies in λ. Then the graded Frobenius characteristic of Mp is Ξλ(t)H1n(x; q, t), a multiple of a Macdonald polynomial. Corresponding results are also given for the span of partial derivatives of an alternant over any complex reflection group. Let (i, j) denote the lattice cell in the i+1st row and j+1st column of the positive quadrant of the plane. If L is a diagram with lattice cells (p1, q1), …, (pn, qn), we set ΔL=det‖xpjiyqji‖ni, j=1, and let ML be the linear span of the partial derivatives of ΔL. The bihomogeneity of ΔL and its alternating nature under the diagonal action of Sn gives ML the structure of a bigraded Sn-module. We give a family of examples and some general conjectures about the bivariate Frobenius characteristic of ML for two dimensional diagrams
Averaging Sequences, Deranged Mappings, and a Problem of Lampert and Slater

AbstractWe answer a question posed by D. E. Lampert and P. J. Slater (1998, Amer. Math. Monthly105, 556–558). Consider a sequence of real numbers qn in the interval [0, 1] defined by q0=0, q1=1, and, for n⩾1, qn+1 equals an average of preceding terms in the sequence. The weights used in the average are provided by a triangular array pn, k of probabilities whose row sums are 1. What is the limiting behavior of a sequence qn so defined? For the Lampert–Slater sequence the weight pn, k is the probability that a randomly chosen fixed-point free mapping of [n+1] omits exactly k elements from its image. To gain some insight into this averaging process, we first analyze what happens with a simpler array of weights pn, k defined in terms of binomial coefficients. One of our theorems states that if the weights pn, k are closely concentrated and the sequence qn exhibits oscillatory behavior up to a certain computable point, then it will exhibit oscillatory behavior from then on. We carry out the computations necessary to verify that the Lampert–Slater sequence satisfies the hypotheses of the latter theorem. A result on martingales (1992, N. Alon, J. H. Spencer, and P. Erdo&#x030B;s, “The Probabilistic Method,” Wiley, New York) is used to prove the close concentration of the weights pn, k
Immanants and Finite Point Processes

AbstractGiven a Hermitian, non-negative definite kernel K and a character χ of the symmetric group on n letters, define the corresponding immanant function Kχ[x1, …, xn]≔∑σχ(σ)∏ni=1K(xi, xσ(i)), where the sum is over all permutations σ of {1, …, n}. When χ is the sign character (resp. the trivial character), then Kχ is a determinant (resp. permanent). The function Kχ is symmetric and non-negative, and, under suitable conditions, is also non-trivial and integrable with respect to the product measure μ⊗n for a given measure μ. In this case, Kχ can be normalised to be a symmetric probability density. The determinantal and permanental cases or this construction correspond to the fermion and boson point processes which have been studied extensively in the literature. The case where K gives rise to an orthogonal projection of L2(μ) onto a finite-dimensional subspace is studied here in detail. The determinantal instance of this special case has a substantial literature because of its role in several problems in mathematical physics, particularly as the distribution of eigenvalues for various models of random matrices. The representation theory of the symmetric group is used to compute the normalisation constant and identify the kth-order marginal densities for 1⩽k⩽n as linear combinations of analogously defined immanantal densities. Connections with inequalities for immanants, particularly the permanental dominance conjecture of Lieb, are considered, and asymptotics when the dimension of the subspace goes to infinity are presented
Left-Modular Elements of Lattices

AbstractLeft-modularity is a concept that generalizes the notion of modularity in lattice theory. In this paper, we give a characterization of left-modular elements and derive two formulae for the characteristic polynomial, χ, of a lattice with such an element, one of which generalizes Stanley's theorem [6] about the partial factorization of χ in a geometric lattice. Both formulae provide us with inductive proofs for Blass and Sagan's theorem [2] about the total factorization of χ in LL lattices. The characteristic polynomials and the Möbius functions of non-crossing partition lattices and shuffle posets are computed as examples
Geometric Identities in Lattice Theory

AbstractAn Arguesian identity is an identity in Grassmann–Cayley algebras with certain multi-linear properties of expressions in joins and meets of vectors and covectors. Many classical theorems of projective geometry and their generalizations to higher dimensions can be expressed as simple and elegant Arguesian identities. In a previous work we showed that an Arguesian identity can be unfolded with respect to a vector variable to obtain a lattice inequality, which holds in various lattices. In this paper, we extend this technique to an arbitrary variable. We prove that for any variable v of an Arguesian identity I, a lattice inequality can be obtained by unfolding I with respect to the variable v. This inequality and its dual are valid in the class of linear lattices if the identity is of order 2, and in the congruence variety of Abelian groups if the identity is of a higher order. Consequently, we obtain a family of lattice identities which are self-dual over the class of linear lattices. In particular, all the inequalities obtained by this method are valid in the lattice of subspaces of a vector space, which are characteristic-free and independent of dimensions
Walks on Directed Graphs and Matrix Polynomials

AbstractWe give a matrix generalization of the family of exponential polynomials in one variable φk(x). Our generalization consists of a matrix of polynomials Φk(X)=(Φ(k)i, j(X))ni, j=1 depending on a matrix of variables X=(xi, j)ni, j=1. We prove some identities of the matrix exponential polynomials which generalize classical identities of the ordinary exponential polynomials. We also introduce matrix generalizations of the decreasing factorial (x)k=x(x−1)(x−2)…(x−k+1), the increasing factorial (x)(k)=x(x+1)(x+2)…(x+k−1), and the Laguerre polynomials. These polynomials have interesting combinatorial interpretations in terms of different kinds of walks on directed graphs
Deformations of Coxeter Hyperplane Arrangements

AbstractWe investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement xi−xj=1, 1⩽i<j⩽n, is equal to the number of alternating trees on n+1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of binary trees, partially ordered sets, and tournaments. More generally, we give formulae for the number of regions and the Poincaré polynomial of certain finite subarrangements of the affine Coxeter arrangement of type An−1. These formulae enable us to prove a “Riemann hypothesis” on the location of zeros of the Poincaré polynomial. We give asymptotics of the Poincaré polynomials when n goes to infinity. We also consider some generic deformations of Coxeter arrangements of type An−1
A Möbius Identity Arising from Modularity in a Matroid Bilinear Form

AbstractThe matrix for the bilinear form of the flag space of a matroid has (with respect to an appropriate basis) a tensor product structure when the matroid has a modular flat K. When determinants are taken, an identity is obtained for the rho function (a certain product of the Möbius and beta functions) summed over flats with a fixed intersection with K. When the identity is interpreted for Dowling lattices and finite projective spaces, identities with similar combinatorial proofs are obtained for binomial and Gaussian coefficients, respectively
Subquadrangles of Generalized Quadrangles of Order (q2, q), q Even

AbstractFirst, we survey the known generalized quadrangles of order (q2, q), q even, including a description of their known subquadrangles of order q. Then, in the case of Tits' generalized quadrangles, we completely classify the subquadrangles of order q, while in the case of the flock quadrangles we classify the subquadrangles of order q which contain the base point. Finally, we determine the full automorphism groups of the Tits generalized quadrangles T2(O) of order q and T3(Ω) of order (q, q2)
On Sequences with Zero Autocorrelation and Orthogonal Designs

AbstractWe propose some methods for multiplying the length and type of sequences with elements on a set of commuting variables which have zero non-periodic autocorrelation function. We use base sequences of lengths n+1, n+1, n, n in order to construct four directed sequences of lengths n+1, n+1, n, n and type (2n+1, 2n+1) with zero NPAF as well as normal sequences of length n in order to construct four directed sequences of length 2n and type (2n, 2n, 2n, 2n) with zero NPAF. We construct two and four directed sequences with zero PAF of length 34 and type (34, 34), and (34, 34, 34, 34), respectively, as well as four directed sequences with zero NPAF of lengths 34, 34, 33, 33 and type (67, 67). We also indicate that from m directed sequences of lengths n1, n2, …, nm which consist of t variables, we obtain k·m directed sequences of lengths n1, n2, …, nm (k sequences will be of lengths ni, i=1, 2, …, m) which consist of k·t variables for k=1, 2, …. The above methods lead to the construction of many new orthogonal designs
Unextendible Product Bases

AbstractLet C denote the complex field. A vector v in the tensor product ⊗mi=1Cki is called a pure product vector if it is a vector of the form v1⊗v2…⊗vm, with vi∈Cki. A set F of pure product vectors is called an unextendible product basis if F consists of orthogonal nonzero vectors, and there is no nonzero pure product vector in ⊗mi=1Cki which is orthogonal to all members of F. The construction of such sets of small cardinality is motivated by a problem in quantum information theory. Here it is shown that the minimum possible cardinality of such a set F is precisely 1+∑mi=1(ki−1) for every sequence of integers k1, k2, …, km⩾2 unless either (i) m=2 and 2∈{k1, k2} or (ii) 1+∑mi=1(ki−1) is odd and at least one ki is even. In each of these two cases, the minimum cardinality of the corresponding F is strictly bigger than 1+∑mi=1(ki−1)
Geometrical Constructions of Flock Generalized Quadrangles

AbstractWith any flock F of the quadratic cone K of PG(3, q) there corresponds a generalized quadrangle S(F) of order (q2, q). For q odd Knarr gave a pure geometrical construction of S(F) starting from F. Recently, Thas found a geometrical construction of S(F) which works for any q. Here we show how, for q odd, one can derive Knarr's construction from Thas' one. To that end we describe an interesting representation of the point-plane flags of PG(3, q), which can be generalized to any dimension and which can be useful for other purposes. Applying this representation onto Thas' model of S(F), another interesting model of S(F) on a hyperbolic cone in PG(6, q) is obtained. In a final section we show how subquadrangles and ovoids of S(F) can be obtained via the description in PG(6, q)
Groups of Maximal Arcs

AbstractApart from hyperovals and their duals there are only three classes of maximal arcs known in Desarguesian projective planes. Two classes are due to J. A. Thas and one to R. H. F. Denniston. In this paper collineation stabiliser and isomorphism problems for those maximal arcs in Desarguesian projective planes are examined. The full collineation stabilisers of the known maximal arcs are calculated, and it is shown that all of one class of Thas maximal arcs and those of the second class of Thas maximal arcs in Desarguesian projective planes arising from elliptic quadrics are isomorphic to those of Denniston. The final result is to classify maximal arcs in Desarguesian projective planes whose collineation stabilisers are transitive on the points of the maximal arcs
A Homotopy Equivalence for Partition Posets Related to Liftings of Sn−1-Modules to Sn

AbstractWe give a homotopy equivalence to explain an Sn−1-module isomorphism which occurs frequently in the homology of subposets of the partition lattice Πn. The isomorphism in question is necessary for the existence of a lifting to Sn of the Sn−1-module involved. It has also been observed in certain deformations of the free Lie algebra. The topological explanation allows us to generate further examples in subposets of Πn. We show that, in these two contexts, it is often accompanied by an Sn-action on the Sn−1-module occurring in the isomorphism. A well-known example is the lifting of the homology of Πn−1
Regular Oberwolfach Problems and Group Sequencings

AbstractWe deal with Oberwolfach factorizations of the complete graphs Kn and K*n, which admit a regular group of automorphisms. We show that the existence of such a factorization is equivalent to the existence of a certain difference sequence defined on the elements of the automorphism group, or to a certain sequencing of the elements of that group. In the particular case of a hamiltonian factorization of the directed graph K*n which admits a regular group of automorphisms G (|G|=n−1), we have that such a factorization exists if and only if G is sequenceable. We shall demonstrate how the mentioned above (difference) sequences may be used in the construction of such factorizations. We prove also that a hamiltonian factorization of the undirected graph Kn (n odd) which admits a regular group of automorphisms G (|G|=(n−1)/2) exists if and only if n≡3 (mod4), without further restrictions on the structure of G
Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

AbstractLet Δ be a finite thick dual polar space of rank 3. We say that a hyperplane H of Δ is locally singular (respectively, quadrangular or ovoidal) if H∩Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Δ. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of Δ or Δ is the dual of Q(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in Q(6, 2) and H(5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry Δ\H induced by Δ on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Δ\H flag-transitive are classified
Cyclic Relative Difference Sets with Classical Parameters

AbstractWe investigate the existence of cyclic relative difference sets with parameters ((qd−1)/(q−1), n, qd−1, qd−2(q−1)/n), q any prime power. One can think of these as “liftings” or “extensions” of the complements of Singer difference sets. When q is odd or d is even, we find that relative difference sets with these parameters exist if and only if n is a divisor of q−1. In case q is even and d is odd, relative difference sets with these parameters exist if and only if n is a divisor of 2(q−1)
Extending Locally Truncated Geometries

AbstractGiven a set of types I, a type 0∈I, a subset J of I containing 0, and a diagram I0 over I\{0}, a geometry Γ over the set of types J is said to be locally truncated of I0-type if the J\{0}-residues of Γ are truncations of geometries or chamber systems belonging to I0. We give a sufficent condition for such a geometry to be the J-truncation of a chamber system over the set of types I with all I\{0}-residues belonging to I0. Then we apply our result to some special cases. We exploit it to classify flag-transitive cn.c*-geometries of rank n+2⩾4 and cn.c*-geometries of order 2. We give a new proof of a theorem of Ronan on Cn.L-geometries and we construct chamber systems for a number of sporadic groups in which certain well known geometries are involved as truncations or residues
A Construction for Infinite Families of Steiner 3-Designs

AbstractLet q be a prime power and a be a positive integer such that a⩾2. Assume that there is a Steiner 3-(a+1, q+1, 1) design. For every v satisfying certain arithmetic conditions we can construct a Steiner 3-(vad+1, q+1, 1) design for every d sufficiently large. In the case of block size 6, when q=5, this theorem yields new infinite families of Steiner 3-designs: if v is a given positive integer satisfying the necessary arithmetic conditions, for every non-negative integer m there exists a Steiner 3-(v(4·5m+1)d+1), 6, 1) for sufficiently large d
Sign-Balanced Posets

AbstractLet P be a finite partially ordered set with a fixed labeling. The sign of a linear extension of P is its sign when viewed as a permutation of the labels of the elements of P. Call P sign-balanced if the number of linear extensions of P of positive sign is the same as the number of linear extensions of P of negative sign. In this paper we determine when the posets in a particular class are sign-balanced. When posets in this class are not sign-balanced, we determine the difference between the number of positive linear extensions and the number of negative linear extensions. One special case of this class is the product of an m-chain with an n-chain, m and n both >1. In this case, we show P is sign-balanced if and only if m≡n mod2