pinecone/core-2020-05-10-deduplication huggingface dataset:
Set Systems with Restricted Intersections modulo Prime Powers

AbstractWe study set systems satisfying Frankl–Wilson-type conditions modulo prime powers. We prove that the size of such set systems is polynomially bounded, in contrast with V. Grolmusz's recent result that for non-prime-power moduli, no polynomial bound exists. More precisely we prove the following result. Theorem.Let p be a prime and q=pk. Let μ1, …, μsbe distinct integers, 0⩽μi⩽q−1. Let X be a set of n elements and let A1, A2, …, Ambe subsets of X with the following properties: •|Ai|≢μℓ (modq) for alli, ℓ, 1⩽i⩽m, 1⩽ℓ⩽s. •For alli, j (1⩽i<j⩽m), there exists ℓ (1⩽ℓ⩽s) such that|Ai∩Aj|≡μℓ(modq).Then[formula]where D ⩽2s−1.D is the minimum degree of polynomials “separating” the set {μ1, …, μs} from the sizes of the Ai modulo q (q=pk). Let D(s, k) be the maximum value of D for fixed s and k (L and p vary). The asymptotic behavior of the function D(s, k) turns out to be rather complicated; we establish polynomially related upper and lower bounds on D(s, k). These bounds give us an idea of the limitations of the method. Our results extend a theorem of Deza, Frankl, and Singhi, who studied the case of prime moduli. The main point we make is that our bound implies a polynomial bound of the form m⩽nc(s) for all prime power moduli q. In this sense the theorem complements a remarkable recent result of Grolmusz that no bound of this form holds for any q which is not a prime power
Small Blocking Sets in Higher Dimensions

AbstractWe show that small blocking sets in PG(n, q) with respect to hyperplanes intersect every hyperplane in 1 modulo p points, where q=ph. The result is then extended to blocking sets with respect to k-dimensional subspaces and, at least when p>2, to intersections with arbitrary subspaces not just hyperplanes. This can also be used to characterize certain non-degenerate blocking sets in higher dimensions. Furthermore we determine the possible sizes of small minimal blocking sets with respect to k-dimensional subspaces
Energy of Convex Sets, Shortest Paths, and Resistance

AbstractLet us assign independent, exponentially distributed random edge lengths to the edges of an undirected graph. Lyons, Pemantle, and Peres (1999, J. Combin. Theory Ser. A86 (1999), 158–168) proved that the expected length of the shortest path between two given nodes is bounded from below by the resistance between these nodes, where the resistance of an edge is the expectation of its length. They remarked that instead of exponentially distributed variables, one could consider random variables with a log-concave tail. We generalize this result in two directions. First, we note that the variables do not have to be independent: it suffices to assume that their joint distribution is log-concave. Second, the inequality can be formulated as follows: the expected length of a shortest path between two given nodes is the expected optimum of a stochastic linear program over a flow polytope, while the resistance is the minimum of a convex quadratic function over this polytope. We show that the inequality between these quantities holds true for an arbitrary polytope provided its blocker has integral vertices
Long Finite Sequences

AbstractLet k be a positive integer. There is a longest finite sequence x1, …, xn in k letters in which no consecutive block xi, …, x2i is a subsequence of any later consecutive block xj, …, x2j. Let n(k) be this longest length. We prove that n(1)=3, n(2)=11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackermann hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3)
A Trace Conjecture and Flag-Transitive Affine Planes

AbstractFor any odd prime power q, all (q2−q+1)th roots of unity clearly lie in the extension field Fq6 of the Galois field Fq of q elements. It is easily shown that none of these roots of unity have trace −2, and the only such roots of trace −3 must be primitive cube roots of unity which do not belong to Fq. Here the trace is taken from Fq6 to Fq. Computer based searching verified that indeed −2 and possibly −3 were the only values omitted from the traces of these roots of unity for all odd q⩽200. In this paper we show that this fact holds for all odd prime powers q. As an application, all odd order three-dimensional flag-transitive affine planes admitting a cyclic transitive action on the line at infinity are enumerated
A Lower Bound for Families of Natarajan Dimension d

AbstractA system F of functions {1, 2, …, n}→{1, 2, …, k} has Natarajan dimension at most d if no (d+1)-element subset A⊂X is 2-shattered. A is 2-shattered if for each x∈A there is a 2-element set Vx⊆{1, 2, …, k} such that for any choice of elements cx∈Vx, a function f∈F exists with f(x)=cx for all x∈A. We improve a lower bound of cdkdnd (due to Haussler and Long) for the maximum size of F of Natarajan dimension at most d by a factor somewhat smaller than k (e.g., by k for d=1). The problem of obtaining a tight bound is related to interesting questions in extremal graph theory
Vector Balancing Games with Aging

AbstractIn this article we study an extension of the vector balancing game investigated by Spencer and Olson (which corresponds to the on-line version of the discrepancy problem for matrices). We assume that decisions in earlier rounds become less and less important as the game continues. For an aging parameter q⩾1 we define the current move to be q times more important than the previous one. We consider two variants of this problem: First, the objective is a balanced partition at the end of the game, and second, it is to ensure a balanced partition throughout the game. We concentrate on the case q⩾2. We give an optimal solution for the first problem and a nearly optimal one for the second
Strongly Regular (α, β)-Geometries

AbstractIn this paper we introduce strongly regular (α, β)-geometries. These are a class of geometries that generalise semipartial geometries. Like semipartial geometries the underlying point graph is strongly regular and this is part of the motivation for studying the geometries. In the paper several necessary conditions for existence are given. Strongly regular (α, β)-reguli are defined, and it is shown how they may be used to construct strongly regular (α, β)-geometries. This generalises similar results by J. A. Thas in (1980, European J. Combin.1, 189–192) constructing semipartial geometries. Several constructions of strongly regular (α, β)-geometries are given, and possible parameters of existence for small cases are listed
Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole

AbstractWe deal with unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a, b+m, c, a+m, b, c+m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (−1)-enumeration of these lozenge tilings. In the case that a=b=c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m=0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det0⩽i, j⩽n−1(ωδij+(m+i+jj)), where ω is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (1979, Invent. Math.53, 193–225), which corresponds to ω=1
Codes over Z2m and Jacobi Forms of Genus n

AbstractThe relations between the complete weight enumerators in genus n of Type II codes over Z2m and Jacobi forms of genus n have been discussed. One derives a map between the invariant spaces of the groups G2m, n (or H2m, n, respectively) and the rings of Jacobi forms (or Siegel modular forms, repectively) of genus n. The MacWilliams identities of the complete weight enumerators in genus n of Type II codes over Z2m imply that the complete weight enumerators in genus n of Type II codes over Z2m is in the invariant space of the groups G2m, n
On the Optimum of Delsarte's Linear Program

AbstractWe are interested in the maximal size A(n, d) of a binary error-correcting code of length n and distance d, or, alternatively, in the best packing of balls of radius (d−1)/2 in the n-dimensional Hamming space. The best known lower bound on A(n, d) is due to Gilbert and Varshamov and is obtained by a covering argument. The best know upper bound is due to McEliece, Rodemich, Rumsey, and Welch, and is obtained using Delsarte's linear programming approach. It is not known whether this is the best possible bound one can obtain from Delsarte's linear program. We show that the optimal upper bound obtainable from Delsarte's linear program will strictly exceed the Gilbert–Varshamov lower bound. In fact, it will be at least as big as the average of the Gilbert–Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound. Similar results hold for constant weight binary codes. The average of the Gilbert–Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound might be the true value of Delsarte's bound. We provide some evidence for this conjecture
Parent-Identifying Codes

AbstractFor a set C of words of length 4 over an alphabet of size n, and for a, b∈C, let D(a, b) be the set of all descendants of a and b, that is, all words x of length 4 where xi∈{ai, bi} for all 1⩽i⩽4. The code C satisfies the Identifiable Parent Property if for any descendant of two code-words one can identify at least one parent. The study of such codes is motivated by questions about schemes that protect against piracy of software. Here we show that for any ε>0, if the alphabet size is n>n0(ε) then the maximum possible cardinality of such a code is less than εn2 and yet it is bigger than n2−ε. This answers a question of Hollmann, van Lint, Linnartz, and Tolhuizen. The proofs combine graph theoretic tools with techniques in additive number theory
Existence of Balanced Simplices on Polytopes

AbstractThe classic Sperner lemma states that in a simplicial subdivision of a simplex in Rn and a labelling rule satisfying some boundary condition there is a completely labeled simplex. In this paper we first generalize the concept of completely labeled simplex to the concept of a balanced simplex. Using this latter concept we then present a general combinatorial theorem, saying that under rather mild boundary conditions on a given labelling function there exists a balanced simplex for any given simplicial subdivision of a polytope. This theorem implies the well-known lemmas of Sperner, Scarf, Shapley, and Garcia as well as some other results as special cases. An even more general result is obtained when the boundary conditions on the labelling function are not required to hold. This latter result includes several results of Freund and Yamamoto as special cases
On Eggs and Translation Generalised Quadrangles

AbstractWe study eggs in PG(4n−1, q). A new model for eggs is presented in which all known examples are given. We calculate the general form of the dual egg for eggs arising from a semifield flock. Applying this to the egg obtained by L. Bader et al. (1999, J. Combin. Theory Ser. A86, 49–62) from the Penttila–Williams ovoid, we obtain the dual egg, which is not isomorphic to any of the previous known examples. Furthermore we give a new proof of a conjecture of J. A. Thas (1994, J. Combin. Theory Ser. A67, 140–160) using our model and classify all eggs of PG(7, 2) which is equivalent to the classification of all translation generalised quadrangles of order (4, 16)
A Four-Class Association Scheme

AbstractWe show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q2), for q⩾4 a power of 2, thereby proving a conjecture of D. de Caen and E. van Dam (Fissioned triangular schemes via the cross-ratio, European J. Combin.22 (2001), 297–301). This is a fusion of certain relations in the fission scheme FT(q2+1) obtained from the triangular association scheme. Combining three relations in the above four-class association scheme yields a strongly regular graph, which we show is isomorphic to one constructed by Brouwer and Wilbrink using hyperbolic solid sections of the parabolic quadric in PG(4, q)
Counting Configurations in Designs

AbstractGiven a t-(v, k, λ) design, form all of the subsets of the set of blocks. Partition this collection of configurations according to isomorphism and consider the cardinalities of the resulting isomorphism classes. Generalizing previous results for regular graphs and Steiner triple systems, we give linear equations relating these cardinalities. For any fixed choice of t and k, the coefficients in these equations can be expressed as functions of v and λ and so depend only on the design's parameters, and not its structure. This provides a characterization of the elements of a generating set for m-line configurations of an arbitrary design
MacMahon Symmetric Functions, the Partition Lattice, and Young Subgroups

AbstractA MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we show that the MacMahon symmetric functions are the generating functions for the orbits of sets of functions indexed by partitions under the diagonal action of a Young subgroup of a symmetric group. We define a MacMahon chromatic symmetric function that generalizes Stanley's chromatic symmetric function. Then, we study some of the properties of this new function through its connection with the noncommutative chromatic symmetric function of Gebhard and Sagan
Two Generalizations of Posets of Shuffles

AbstractWe study posets defined by Stanley as a multiset generalization of Greene's posets of shuffles. Ehrenborg defined a quasi-symmetric function encoding for the flag f-vector, denoted FP, and we determine FP for shuffle posets of multisets, expressing it as a Schur-positive symmetric function. This leads to several combinatorial formulas as well as proofs that shuffle posets of multisets are supersolvable and have symmetric chain decompositions. We also generalize posets of shuffles to posets for shuffling k words, answering a question of Stanley. Finally, we extend our results about shuffle posets of multisets to k-shuffle posets
A Theory of Ternary Complementary Pairs

AbstractSequences with zero autocorrelation are of interest because of their use in constructing orthogonal matrices and because of applications in signal processing, range finding devices, and spectroscopy. Golay sequences, which are pairs of binary sequences (i.e., all entries are ±1) with zero autocorrelation, have been studied extensively, yet are known only in lengths 2a10b26c. Ternary complementary pairs are pairs of (0, ±1)-sequences with zero autocorrelation (thus, Golay pairs are ternary complementary pairs with no 0's). Other kinds of pairs of sequences with zero autocorrelation, such as those admitting complex units for nonzero entries, are studied in similar contexts. Work on ternary complementary pairs is scattered throughout the combinatorics and engineering literature where the majority approach has been to classify pairs first by length and then by deficiency (the number of 0's in a pair); however, we adopt a more natural classification, first by weight (the number of nonzero entries) and then by length. We use this perspective to redevelop the basic theory of ternary complementary pairs, showing how to construct all known pairs from a handful of initial pairs we call primitive. We display all primitive pairs up to length 14, more than doubling the number that could be inferred from the existing literature
A Characterization of Exponential and Ordinary Generating Functions

AbstractIt is common to represent a sequence a=(a0, a1, …) of complex numbers with a generating function. G.C. Rota once remarked that among all the possible generating functions that might be used to represent a, the ordinary and exponential generating functions are the most ubiquitous. It is unclear what, if anything, makes these two particular representations special. We show here that the ordinary and exponential representations uniquely possess the property that the determinants of the Hankel matrices of certain convolutional polynomials in a are independent of a1. Hankel matrices are closely associated with the problem of moments and the problem of moment preserving maps and hence the independence of a1 has some curious implications. For example determining if a is a sequence of cumulants for some distribution is necessarily independent of the value for the mean a1. We explore this and other applications of the generating function characterization property within probability and combinatorics
The Geometry of Frequency Squares

AbstractThis paper establishes a correspondence between mutually orthogonal frequency squares (MOFS) and nets satisfying an extra property (“framed nets”). In particular, we provide a new proof for the bound on the maximal size of a set of MOFS and obtain a geometric characterisation of the case of equality: necessary and sufficient conditions for the existence of a complete set of MOFS are given in terms of the existence of a certain type of PBIBD based on the L2-association scheme. We also discuss examples obtained from classical affine geometry and recursive construction methods for (complete) sets of MOFS
Partitioning the Boolean Lattice into Chains of Large Minimum Size

AbstractLet 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1, …, n} ordered by inclusion. Recall that 2[n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of Füredi, we show that there exists a function d(n)∼12n such that for any n⩾0, 2[n] may be partitioned into (n⌊n/2⌋) chains of size at least d(n). (For comparison, a positive answer to Füredi's question would imply that the same result holds for some d(n)∼π/2n.) More precisely, we first show that for 0⩽j⩽n, the union of the lowest j+1 elements from each of the chains in the CSCD of 2[n] forms a poset Tj(n) with the normalized matching property and log-concave rank numbers. We then use our results on Tj(n) to show that the nodes in the CSCD chains of size less than 2d(n) may be repartitioned into chains of large minimum size, as desired
The Distributions of the Entries of Young Tableaux

AbstractLet T be a standard Young tableau of shape λ⊢k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n→∞, equal to fλ/k!, where fλ is the number of all tableaux of shape λ. In other words, the probability that a large tableau contains T is equal to the number of tableaux whose shape is that of T, divided by k!. We give several applications, to the probabilities that a set of prescribed entries will appear in a set of prescribed cells of a tableau, and to the probabilities that subtableaux of given shapes will occur. Our argument rests on a notion of quasirandomness of families of permutations, and we give sufficient conditions for this to hold
On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body

AbstractA Minkowski space Md=(Rd, ‖‖) is just Rd with distances measured using a norm ‖‖. A norm ‖‖ is completely determined by its unit ball {x∈Rd∣‖x‖⩽1} which is a centrally symmetric convex body of the d-dimensional Euclidean space Ed. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in Md, d⩾3. In fact, we deal with a somewhat more general problem namely, we give upper bounds for the maximum number of touching pairs in a packing of n translates of a given convex body in Ed, d⩾3
Compactness Theorems for Geometric Packings

AbstractMoser asked whether the collection of rectangles of dimensions 1×12, 12×13, 13×14, …, whose total area equals 1, can be packed into the unit square without overlap, and whether the collection of squares of side lengths 12, 13, 14, … can be packed without overlap into a rectangle of area π2/6−1. Computational investigations have been made into packing these collections into squares of side length 1+ε and rectangles of area π2/6−1+ε, respectively, and one can consider the apparently weaker question of whether such packings are possible for every positive number ε. In this paper we establish a general theorem on sequences of geometrical packings that implies, in particular, that the “for every ε” versions of these two problems are actually equivalent to the original tiling problems
A General Construction for Optimal Cyclic Packing Designs

AbstractCyclic packing designs of pairs have various applications in communications. In this paper, the concept of a (g1, g2, …, gr;u)-regular cyclic packing design is defined, and used to establish a quite general recursive construction concerning cyclic packing designs. As corollaries, we are able to unify many known constructions for cyclic designs. As an application, we obtain infinite series of new optimal cyclic packing designs which can be utilized directly to produce new optimal optical orthogonal codes
Random Walk in an Alcove of an Affine Weyl Group, and Non-colliding Random Walks on an Interval

AbstractWe use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walks in the alcoves of the classical affine Weyl groups. In all cases, we get determinant formulas for the number of k-step walks. One important example is the region m>x1>x2>…>xn>0, which is a rescaled alcove of the affine Weyl group Cn. If each coordinate is considered to be an independent particle, this models n non-colliding random walks on the interval (0, m). Another case models n non-colliding random walks on a circle
Hypergraphs, Quasi-randomness, and Conditions for Regularity

AbstractHaviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. They proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this paper, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0<d<1) and prove that the condition of having asymptotically vanishing discrepancy is equivalent to several other quasi-random properties of H, similar to the ones introduced by Chung and Graham. In particular, we prove that the correct “spectrum” of the s-vertex subhypergraphs is equivalent to quasi-randomness for any s⩾2k. Our work may be viewed as a continuation of the work of Chung and Graham, although our proof techniques are different in certain important parts
New Maximal Arcs in Desarguesian Planes

AbstractConstructions are described of maximal arcs in Desarguesian projective planes utilizing sets of conics on a common nucleus in PG(2, q). Several new infinite families of maximal arcs in PG(2, q) are presented and a complete enumeration is carried out for Desarguesian planes of order 16, 32, and 64. For each arc we list the order of its stabilizer and the numbers of subarcs it contains. Maximal arcs may be used to construct interesting new partial geometries, 2-weight codes, and resolvable Steiner 2-designs