pinecone/core-2020-05-10-deduplication huggingface dataset:
A Matroid Generalization of a Result on Row-Latin Rectangles

AbstractLet A be an m×n matrix in which the entries of each row are all distinct. A. A. Drisko (1998, J. Combin. Theory Ser. A84, 181–195) showed that if m⩾2n−1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in the ground set of a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Finally, we observe that m⩾2n−1 appears to force the existence of not merely one but many transversals. We discuss a number of conjectures related to this observation (some of which involve matroids and some of which do not)
On the Number of Permutations Avoiding a Given Pattern

AbstractLet σ∈Sk and τ∈Sn be permutations. We say τ contains σ if there exist 1⩽x1<x2<…<xk⩽n such that τ(xi)<τ(xj) if and only if σ(i)<σ(j). If τ does not contain σ we say τ avoids σ. Let F(n, σ)=|{τ∈Sn∣τavoidsσ}|. Stanley and Wilf conjectured that for any σ∈Sk there exists a constant c=c(σ) such that F(n, σ)⩽cn for all n. Here we prove the following weaker statement: For every fixed σ∈Sk, F(n, σ)⩽cnγ*(n), where c=c(σ) and γ*(n) is an extremely slow growing function, related to the Ackermann hierarchy
Sets of Integers with Missing Differences

AbstractThis paper deals with the problem of finding the maximal density, μ(M), of sets of integers in which differences given by a set M do not occur. The problem is solved for the case where the elements of M are in arithmetic progression. Besides finding lower bounds for most members of the general three element set M, μ(M) has been found for most members of the family {i, j, k}, where i/d≡j/d (mod2) and gcd(i, j)=d. For i/d≢j/d (mod2) and gcd(i, j)=d it is conjectured that the lower bound found is the best possible. A lower bound is given for μ(M) for the set M={i, j, i+j} and μ(M) for certain infinite families of four element set M have been found
Decompositions of Edge-Colored Complete Graphs

AbstractWe prove an asymptotic existence theorem for decompositions of edge-colored complete graphs into prespecified edge-colored subgraphs. Many combinatorial design problems fall within this framework. Applications of our main theorem require calculations involving the numbers of edges of each color and degrees of each color class of edges for the graphs allowed in the decomposition. We do these calculations to provide new proofs of the asymptotic existence of resolvable designs, near resolvable designs, group divisible designs, and grid designs. Two further applications are the asymptotic existence of skew Room d-cubes and the asymptotic existence of (v, k, 1)-BIBDs with any group of order k−1 as an automorphism group
The Distribution of the Maximum Vertex Degree in Random Planar Maps

AbstractWe determine the limiting distribution of the maximum vertex degree Δn in a random triangulation of an n-gon, and show that it is the same as that of the maximum of n independent identically distributed random variables G2, where G2 is the sum of two independent geometric(1/2) random variables. This answers affirmatively a question of Devroye, Flajolet, Hurtado, Noy and Steiger, who gave much weaker almost sure bounds on Δn. An interesting consequence of this is that the asymptotic probability that a random triangulation has a unique vertex with maximum degree is about 0.72. We also give an analogous result for random planar maps in general
Linear Inequalities for Flags in Graded Partially Ordered Sets

AbstractThe closure of the convex cone generated by all flag f-vectors of graded partially ordered sets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of partially ordered sets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded partially ordered sets of rank at most 5
Balancing Unit Vectors

AbstractTheoremA.Letx1,…,x2k+1be unit vectors in a normed plane. Then there exist signsε1,…,ε2k+1∈{±1} such that ‖∑2k+1i=1εixi‖⩽1. We use the method of proof of the above theorem to show the following point facility location result, generalizing Proposition 6.4 of Y. S. Kupitz and H. Martini (1997). TheoremB.Letp0,p1,…,pnbe distinct points in a normed plane such that for any 1⩽i<j⩽nthe closed angle ∠pip0pjcontains a ray opposite some[formula], 1⩽k⩽n. Thenp0is a Fermat–Torricelli point of {p0,p1,…,pn}, i.e.x=p0minimizes ∑ni=0‖x−pi‖. We also prove the following dynamic version of Theorem A. TheoremC.Letx1,x2,… be a sequence of unit vectors in a normed plane. Then there exist signsε1,ε2,…∈{±1} such that ‖∑2ki=1εixi‖⩽2 for allk∈N. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C
Weighted-Inversion Statistics and Their Symmetry Groups

AbstractA statistic w on Sn is a weighted-inversion (w-i) statistic if there exist weights wi, j such that w(σ)=∑i<jχ[σi>σj]wi, j for each σ∈Sn. Two well-known examples are the major index and inversion count statistics. These two statistics share the same distribution over Sn, and many bijections Sn→Sn have been described to prove this. These bijections thus have the property that they map a certain w-i statistic to another. This paper presents the results of our search for bijections φ: Sn→Sn with an even stronger property: given any w-i statistic w, the statistic w∘φ is also a w-i statistic. Such a set of bijections forms a group, which we call the core group of Sn. We exhibit a subgroup of the core group of Sn which is isomorphic to the dihedral group Dn+1. We extend these ideas to other sets of objects, including subsets of Sn and sets of permutations of a multiset. As examples, we develop a family of subsets of Sn which has a core group isomorphic to a Weyl group of order 2n·n!, and we show that the set of permutations of the multiset {0k, 1n−k} has a core group containing Sk×Sn−k as a subgroup. We demonstrate that the core group of a set A is the group of permutations of the rows of a certain matrix H (depending only on the inversion patterns of the objects in A) which preserve the column space of H. This allows us to compute the core group with no knowledge of the actual w-i statistics involved
On the Universal Embedding of Dual Polar Spaces of Type Sp2n(2)

AbstractIt has been conjectured by A. E. Brouwer that the dimension of the universal embedding module of a dual polar space of type Sp2n(2) is λ(n)=(2n+1)(2n−1+1)/3. Following a point stabilizer approach of A. A. Ivanov and M. K. Bardoe, we investigate the dimensions of certain quotients of permutation modules for SLn(2) on subspaces of a fixed vector space of dimension n. This is accomplished by studying the nullity of associated incidence matrices over GF(2). In the process we provide evidence of a generating set for the dual polar space of type Sp2n(2) of cardinality λ(n)
Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 1

AbstractThe flag geometry Γ=(P, L, I) of a finite projective plane Π of order s is the generalized hexagon of order (s, 1) obtained from Π by putting P equal to the set of all flags of Π, by putting L equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). Then we say that Γ is fully and weakly embedded in the finite projective space PG(d, q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of Γ generates PG(d, q), and if the set of points of Γ not opposite any given point of Γ does not generate PG(d, q). In an earlier paper, we have shown that the dimension d of the projective space belongs to {6, 7, 8}, and that the projective plane Π is Desarguesian. Furthermore, we have given examples for d=6, 7. In the present paper we show that for d=6, only these examples exist, and we also partly handle the case d=7. More precisely, we completely classify the full and weak embeddings of Γ (Γ as above) in the case that there are two opposite lines L, M of Γ with the property that the subspace of PG(d, q) generated by all lines of Γ meeting either L or M has dimension 6 (which is the case for all embeddings in PG(d, q), d∈{6, 7}). Together with Parts 2 and 3, this will provide the complete classification of all full and weak embeddings of Γ
Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 2

AbstractThe flag geometry Γ=(P,L,I) of a finite projective plane Π of order s is the generalized hexagon of order (s,1) obtained from Π by putting P equal to the set of all flags of Π, by putting L equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). Then we say that Γ is fully and weakly embedded in the finite projective space PG(d,q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d,q), if s=q, if the set of points of Γ generates PG(d,q), and if the set of points of Γ not opposite any given point of Γ does not generate PG(d,q). In two earlier papers we have shown that the dimension d of the projective space belongs to {6, 7, 8}, that the projective plane Π is Desarguesian, and we have classified the full and weak embeddings of Γ (Γ as above) in the case that there are two opposite lines L, M of Γ with the property that the subspace UL,M of PG(d,q) generated by all lines of Γ meeting either L or M has dimension 6 (which is automatically satisfied if d=6). In the present paper, we partly handle the case d=7; more precisely, we consider for d=7 the case where for all pairs (L,M) of opposite lines of Γ, the subspace UL,M has dimension 7 and where there exist four lines concurrent with L contained in a 4-dimensional subspace of PG(7,q)
Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 3

AbstractThe flag geometry Γ=(P, L, I) of a finite projective plane Π of order s is the generalized hexagon of order (s, 1) obtained from Π by putting P equal to the set of all flags of Π, by putting L equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). Then we say that Γ is fully and weakly embedded in the finite projective space PG(d, q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of Γ generates PG(d, q), and if the set of points of Γ not opposite any given point of Γ does not generate PG(d, q). In three earlier papers we have shown that the dimension d of the projective space belongs to {6, 7, 8}, that the projective plane Π is Desarguesian, and we have classified the full and weak embeddings of Γ (Γ as above) for d=6 and for d=7 in the case that there exists a line L of Γ and four distinct lines L1, L2, L3, L4 concurrent with Γ which generate a 4-dimensional space. In the present paper, we drop all these additional assumptions by completing the case d=7 and handling the case d=8. In particular, we find new examples for d=8 (contrary to our original conjecture (J. A. Thas and H. Van Maldeghem, Des. Codes Cryptogr.17 (1999), 97–104)). This means that we have now the complete classification of all fully and weakly embedded geometries Γ in PG(d, q), with Γ the flag geometry of a finite projective plane
Truncated Transversal Designs: A New Lower Bound on the Number of Idempotent MOLS of Side n

AbstractA truncated transversal design TTD of type gkm1 is a {k,k+1}-GDD of type gkm1 in which each point on the group of size m lies only in blocks of size k+1. Thus a TTD of type gkm1 is equivalent to a transversal design TD (k,g) having m disjoint parallel classes of blocks. We employ a new construction developed by the author (1993, J. Combin. Des.1, 15–26) to show that if g1<g2 and if there exists a TD (k,g1) and a TD (k+1,g2), then there exists a TTD of type (g1g2)km1 for any 0⩽m⩽(g2divg1)g21. As a corollary, we obtain a new lower bound on the number of mutually orthogonal idempotent latin squares of side g: if g1<g2 and there exist r MOLS of side g1 and r+1 MOLS of side g2 , then N(1 g1g2)⩾r
Finite-Ring Combinatorics and MacWilliams' Equivalence Theorem

AbstractF. J. MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been genera- lized by J. Wood who proved it for Frobenius rings using character theoretic methods. The present paper provides a combinatorial approach: First we extend I. Constantinescu's concept of homogeneous weights on arbitrary finite rings and prove MacWilliams' equivalence theorem to hold with respect to these weights for all finite Frobenius rings. As a central tool we then establish a general inversion principle for real functions on finite modules that involves Möbius inversion on partially ordered sets. An application of the latter yields the aforementioned result of Wood
Simultaneous Approximations and Covering by Arithmetic Progressions in Fp

AbstractGiven a set A={a1, …, an}⊆Fp of residues modulo prime p, we seek α, δ∈Fp (δ≠0) which simultaneously minimize all the distances ‖δai−α‖ from the zero residue and investigate the quantity mn=max|A|=nminα, δmax1⩽i⩽n‖δai−α‖, the outer maximum being taken over all n-element subsets of Fp. It is shown that this extremal simultaneous approximation problem is equivalent to the combinatorial problem of finding minimal ln such that any set of n residues modulo p can be covered by an arithmetic progression of the length ln. For n⩾4, we determine the order of magnitude of mn and prove that 12p1−1/(n−1)(1+o(1))<mn<n−1/(n−1)p1−1/(n−1)(1+o(1)) (as p→∞ and n is small compared to p). For n=3, we find a sharp asymptotic and moreover, prove that −p/3<m3−p/3<1/2. These results answer a question of Straus about the maximum possible affine diameter of an n-element set of residues modulo a prime
Intersecting Designs

AbstractWe prove the intersection conjecture for designs: For any complete graph Kr there is a finite set of positive integers M(r) such that for every n>n0(r), if Kn has a Kr-decomposition (namely a 2-(n, r, 1) design exists) then there are two Kr-decompositions of Kn having exactly q copies of Kr in common for every q belonging to the set[formula]. In fact, this result is a special case of a much more general result, which determines the existence of k distinct Kr-decompositions of Kn which have q elements in common, and all other elements of any two of the decompositions share at most one edge in common
Existence of Z-Cyclic Triplewhist Tournaments for a Prime Number of Players

AbstractA Z-cyclic triplewhist tournament for 4n+1 players, or briefly a TWh(4n+1), is equivalent to a n-set {(ai, bi, ci, di) | i=1,…,n} of quadruples partitioning Z4n+1−{0} with the property that ∪ni=1{±(ai−ci), ±(bi−di)}=∪ni=1{±(ai−bi), ±(ci−di)}=∪ni=1{±(ai−di), ±(bi−ci)}=Z4n+1−{0}. The existence problem for Z-cyclic TWh(p)'s with p a prime has been solved for p≢1 (mod16). I. Anderson et al. (1995, Discrete Math.138, 31–41) treated the case of p≡5 (mod8) while Y. S. Liaw (1996, J. Combin. Des.4, 219–233) and G. McNay (1996, Utilitas Math.49, 191–201) treated the case of p≡9 (mod16). In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes p≡1 (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all≡1 (mod4) and distinct from 5, 13, and 17
Maximum Antichains in Random Subsets of a Finite Set

AbstractWe consider the random poset P(n,p) which is generated by first selecting each subset of [n]={1,…,n} with probability p and then ordering the selected subsets by inclusion. We give asymptotic estimates of the size of the maximum antichain for arbitrary p=p(n). In particular, we prove that if pn/logn→∞, an analogue of Sperner's theorem holds: almost surely the maximum antichain is (to first order) no larger than the antichain which is obtained by selecting all elements of P(n,p) with cardinality ⌊n/2⌋. This is almost surely not the case if pn=↛∞
Posets That Locally Resemble Distributive Lattices An Extension of Stanley's Theorem (with Connections to Buildings and Diagram Geometries)

AbstractLet P be a graded poset with 0 and 1 and rank at least 3. Assume that every rank 3 interval is a distributive lattice and that, for every interval of rank at least 4, the interval minus its endpoints is connected. It is shown that P is a distributive lattice, thus resolving an issue raised by Stanley. Similar theorems are proven for semimodular, modular, and complemented modular lattices. As a corollary, a theorem of Stanley for Boolean lattices is obtained, as well as a theorem of Grabiner (conjectured by Stanley) for products of chains. Applications to incidence geometry and connections with the theory of buildings are discussed
Perfect Hash Families: Probabilistic Methods and Explicit Constructions

AbstractAn (n, q, t)-perfect hash family of size s consists of a set V of order n, a set F of order q, and a sequence φ1, φ2, …, φs of functions from V to F with the following property. For all t-subsets X⊆V, there exists i∈{1, 2, …, s} such that φi is injective when restricted to X. An (n, q, t)-perfect hash family of minimal size is known as optimal. The paper presents a probabilistic existence result for perfect hash families which improves on the well known result of Mehlhorn for many parameter sets. The probabilistic methods are strong enough to establish the size of an optimal perfect hash family in many cases. The paper also gives several explicit constructions of classes of perfect hash families
The 2-Adic Behavior of the Number of Partitions into Distinct Parts

AbstractLet Q(n) denote the number of partitions of an integer n into distinct parts. For positive integers j, the first author and B. Gordon proved that Q(n) is a multiple of 2j for every non-negative integer n outside a set with density zero. Here we show that if i≢0 (mod2j), then#{0⩽n⩽X:Q(n)≡i(mod2j)}⪢jX/logX.In particular, Q(n) lies in every residue class modulo 2j infinitely often. In addition, we examine the behavior of Q(n) (mod8) in detail, and we obtain a simple “closed formula” using the arithmetic of the ring Z[−6]
Perfect Square Packings

AbstractAn interesting problem is to determine whether all the squares of side n−1 can be packed into a rectangle of the appropriate area. Such a packing (into a rectangle of the right area) is called perfect. In this paper, we define an algorithm based on an algorithm by Paulhus and use it to show that there is a perfect packing of the squares of side n−3/5 into a square. The technique can be used to prove that there is a perfect packing of the squares of side n−t into a square, where 1/2<t⩽3/5, provided a certain algorithm succeeds for that value of t. It has succeeded for every such value of t that the author has tried. We also show that there is a perfect packing for all t in the range 0.5964⩽t⩽0.6
Bounds on the Number of Affine, Symmetric, and Hadamard Designs and Matrices

AbstractLower bouds on the number of non-isomorphic embeddings of a symmetric net into affine designs with classical parameters, of an affine design into symmetric designs with classical parameters, and of a symmetric Hadamard design of order n into ones of order 2n are obtained. The bound of Jungnickel on the number of affine 2-(qd, qd−1, (qd−1−1)/(q−1)) designs (d⩾3) that contain the classical (q, qd−2)-net is improved by a factor of q3+4+…+d(q−1)d−2. Similarly, the bound of Jungnickel for the number of symmetric 2-((qd+1−1)/(q−1), (qd−1)/(q−1), (qd−1−1)/(q−1)) designs (d⩾3) that contain the the classical affine design AG(d, q) as a residual design is improved to match that of Kantor. Furthermore, for d large and by starting with rigid symmetric and affine designs, the lower bound for the number of non-isomorphic symmetric 2-((qd+1−1)/(q−1), (qd−1)/(q−1), (qd−1−1)/(q−1)) designs is improved to (qd−1+…+q)!. By using the Paley design of order n=(q+1)/4, q≡3 (mod4) a prime power, a lower bound for the number of Hadamard designs of order q+1 is also obtained. In particular, by choosing a non-classical net and non-classical affine design as the starting point, the bound on the number of symmetric 2-(40, 13, 4) designs is improved from 389 to 1, 108, 800, and the bound on the number of affine 2-(64, 16, 5) designs is improved from 157 to 10, 810, 800. A similar method also improves the number of non-isomorphic Hadamard 2-(31, 15, 7) designs from 1, 266, 891 to 11, 727, 788 and the number of non-isomorphic Hadamard 2-(39, 19, 9) designs from 38 to 5.87×1014. The number of inequivalent Hadamard matrices of order 40 is at least 3.66×1011
Partition Regular Structures Contained in Large Sets Are Abundant

AbstractFurstenberg and Glasner have shown that for a particular notion of largeness in a group, namely piecewise syndeticity, if a set B is a large subset Z, then for any l∈N, the set of length l arithmetic progressions lying entirely in B is large among the set of all length l aritmetic progressions. We extend this result to apply to infinitely many notions of largeness in arbitrary semigroups and to partition regular structures other than arithmetic progressions. We obtain, for example, similar results for the Hales–Jewett theorem
Cocyclic Orthogonal Designs and the Asymptotic Existence of Cocyclic Hadamard Matrices and Maximal Size Relative Difference Sets with Forbidden Subgroup of Size 2

AbstractThis paper contains a discussion of cocyclic Hadamard matrices, their associated relative difference sets, and regular group actions. Nearly all central extensions of the elementary abelian 2-groups by Z2 are shown to act regularly on the associated group divisible design of the Sylvester Hadamard matrices. Cocyclic orthogonal designs are then introduced, and the construction and classification questions for cocyclic M-concordant systems of orthogonal designs are addressed. (M-concordance generalises the concepts of amicability and anti-amicability.) We give an algebraic procedure for constructing and classifying these designs when each indeterminate is constrained to appear just once in each row and column of the orthogonal designs. This paper also gives a general (but apparently not comprehensive) method for constructing systems with no zero entries. In particular, we obtain a cocyclic pair of amicable OD(16; 8, 8). Using this pair of designs, we prove there is a cocyclic Hadamard matrix of order 2ts for any odd integer s>1 and any t⩾⌊8log2s⌋. A consequence of our argument is the theorem, “Let S be any group of odd order s containing k prime order subgroups Si of S such that (1) for i=0, …, k−1, the sets Ui=Si+1Si+2…Sk are subgroups of S, (2) for i=0, …, k−1, Si∩Ui=〈1〉, and (3) U0=S. Let T be any group of order 2t+1 containing a central involution z such that (1) T/〈z〉 is elementary abelian, and (2) the largest elementary abelian direct factor of T has rank between 4log2s and t−2−4log2s. Then there is a normal relative difference set of size 2ts with forbidden subgroup 〈z〉 in the group T×S.” The condition on S holds for any group of odd square-free order or any direct product of odd order elementary abelian groups