pinecone/core-2020-05-10-deduplication huggingface dataset:
Combinatorial Construction of Some Near Polygons

AbstractWe give a construction which takes a rank two incidence geometry with three points on a line and returns a geometry of the same type, i.e., with three points on a line. It is also demonstrated that embeddings of the original geometry can be extended to the new geometry. It is shown that the family of dual polar spaces of type Sp(2n, 2) arise recursively from the construction starting with the geometry consisting of one point and no lines. Making use of this construction we inductively construct projective embeddings for these geometries, in particular the embedding in the spin module for the group Sp(2n, 2). We also show that if we apply the construction to a classical near polygon which is isometrically embedded in the near 2n-gon of type Sp(2n, 2) the resulting space is a near polygon. Examples of such classical, isometrically embedding spaces are near 2n-gon of Hamming type on a three letter alphabet and the product of dual polar spaces of types Sp(2k, 2) and Sp(2l, 2) withk+l=n
Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

AbstractIn the standard Coxeter presentation, the symmetric groupSnis generated by the adjacent transpositions (1, 2), (2, 3), …, (n−1, n). For any given permutation, we consider all minimal-length factorizations thereof as a product of the generators. Any two transpositions (i, i+1) and (j, j+1) commute if the numbersiandjare not consecutive; thus, in any factorization, their order can be switched to obtain another factorization of the same permutation. Extending this to an equivalence relation, we establish a bijection between the resulting equivalence classes and rhombic tilings of a certain 2n-gon determined by the permutation. We also study the graph structure induced on the set of tilings by the other Coxeter relations. For a special case, we use lattice-path diagrams to prove an enumerative conjecture by Kuperberg and Propp, as well as aq-analogue thereof. Finally, we give similar constructions for two other families of finite Coxeter groups, namely those of typesBandD
Construction and Representation of Neighborly Manifolds

AbstractWe deal with neighborly 2-manifolds and 2-pseudomanifolds. We describe a representation of these objects, which is independent of the particular labelling of their vertices. This representation also enables an easy calculation of the automorphism group. We describe two operations which, when applied to one such object, enable us to obtain many (and perhaps all) other such objects with the same number of vertices. Finally, we apply those operations to manifolds and pseudomanifolds withvvertices, for some small values ofv
On Dense Bipartite Graphs of Girth Eight and Upper Bounds for Certain Configurations in Planar Point–Line Systems

AbstractIn earlier work we showed that ifG(m, n) is a bipartite graph with no 4-cycles or 6-cycles, and ifm<c1n2andn<c2m2, then the number of edgeseisO((mn)2/3). Here we give a more streamlined proof, obtaining some sharp results; for example, ifGhas minimum degree at least two thene⩽2 (mn)2/3, and this is a tight bound. Furthermore, one may allowO(mn) 6-cycles and still obtaine=O((mn)2/3). This leads us to conjecture that, ifG(m, n) is the point–line incidence graph of anynpoints andmlines in the plane, then the number of 6-cycles isO(mn). The main result of this paper is a proof that the number of 3-paths in such a graph isO(mn); this is related to the above conjecture
Probabilistic Bounds on the Coefficients of Polynomials with Only Real Zeros

AbstractThe work of Harper and subsequent authors has shown that finite sequences (a0, …, an) arising from combinatorial problems are often such that the polynomialA(z):=∑nk=0akzkhas only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming theakare nonnegative,A(1)>0 and thatA(z) is not constant, it is known thatA(z) has only real zeros iff the normalized sequence (a0/A(1), …, an/A(1)) is the probability distribution of the number of successes innindependent trials for some sequence of success probabilities. Such sequences (a0, …, an) are also known to be characterized by total positivity of the infinite matrix (ai−j) indexed by nonnegative integersiandj. This papers reviews inequalities and approximations for such sequences, calledPólya frequency sequenceswhich follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates
New Maximal Two-Distance Sets

AbstractA two-distance set in Edis a point setXin thed-dimensional Euclidean space such that the distances between distinct points inXassume only two different non-zero values. Based on results from classical distance geometry, we develop an algorithm to classify, for a givend, all maximal (largest possible) two-distance sets in Ed. Using this algorithm we have completed the full classification for alld⩽7, and we have found one set in E8whose maximality follows from Blokhuis' upper bound on sizes ofs-distance sets. While in the dimensionsd⩽6 our classifications confirm the maximality of previously known sets, the results in E7and E8are new. Their counterpart in dimensiond⩾10 is a set of unit vectors with only two values of inner products in the Lorentz space Rd, 1. The maximality of this set again follows from a bound due to Blokhuis
Fractional v. Integral Covers in Hypergraphs of Bounded Edge Size

AbstractIn the early 1980's, V. Rödl proved the Erdős–Hanani Conjecture, sparking a remarkable sequence of developments in the theory of packing and covering in hypergraphs of bounded edge size. Generalizations were given by P. Frankl and Rödl, by N. Pippenger, and by others. In each case, an appropriatesemi-randommethod was used to “construct” the desired optimal object (covering, matching, colouring) in several random stages, followed by a greedy stage. The current work, which further generalizes some of the above results, is again probabilistic, and uses, in addition to earlier ideas, connections with so-calledhard-coredistributions on the set of matchings of a graph. For fixedk⩾2, H ak-bounded hypergraph, andt:H→R+a fractional cover, a sufficient condition is given to ensure that the edge cover numberρ(H), i.e., the size of a smallest set of edges of H with unionV(H), is asymptotically at mostt(H)=∑A∈Ht(A). This settles a conjecture first publicized in Visegrád, June 199
On the Möbius Number of the Subgroup Lattice of the Symmetric Group

AbstractIn this paper, a formula is given for the Möbius numberμ(1, Sn) of the subgroup lattice of the symmetric groupSn. This formula involves the Möbius numbers of certain transitive subgroups ofSn. Whennhas at most two (not necessarily distinct) prime factors ornis a power of two, this formula is refined so that it involves only the Möbius numbers of certain primitive subgroups ofSn. Using the O'Nan-Scott Theorem, the classification of finite simple groups, and the refined formula, the exact value ofμ(1, Sn) is determined whennis prime, twice a prime, or a power of two. For certain primesp, |μ(1, S2p)|≠(2p)!/2. This result gives a negative answer to a question raised by Stanley
Limiting Distributions for the Number of Distinct Component Sizes in Relational Structures

AbstractWe investigate from probabilistic point of view the asymptotic behavior of the number of distinct component sizes in general classes of combinatorial structures of sizenasn→∞. Mild restrictions of admissibility type are imposed on the corresponding generating functions and asymptotic expressions of the mean and variance of that number are obtained. Then we establish weak convergence to a convolution of two distributions, where one of them is always Gaussian. As an illustration we consider three typical generating function examples: partitions of a finite set, partitions of a positive integer and mappings of a finite set into itself
A Robinson–Schensted Algorithm for a Class of Partial Orders

AbstractLetPbe a finite partial order which does not contain an induced subposet isomorphic with 3+1, and letGbe the incomparability graph ofP. Gasharov has shown that the chromatic symmetric functionXGhas nonnegative coefficients when expanded in terms of Schur functions; his proof uses the dual Jacobi–Trudi identity and a sign-reversing involution to interpret these coefficients as numbers ofP-tableau. This suggests the possibility of a direct bijective proof of this result, generalizing the Robinson–Schensted correspondence. We provide such an algorithm here under the additional hypothesis thatPdoes not contain an induced subposet isomorphic with {x>a<b<c>y}
Linear Spaces and Partitioning the Projective Plane

AbstractThe aim of this paper is to settle a question about the partitioning of the projective plane by lines except for a small set. Suppose thatQis a set of points in the projective plane of ordernandΠis a set of lines that partitions the complement ofQ. IfQhas at most 2n−1 points andPhas less thann+1+nlines, then these lines are concurrent. An example is given which shows that the condition on the number of points ofQis sharp. However, it turns out that this is a ‘pathological’ example and if we exclude this case, then the statement can be improved
Urn Sampling and a Majorization Inequality

AbstractIn the context of a certain urn-sampling game, Bennett has studied pairs of sequences for which the products of successive finite differences of the sequences are majorized by the differences of the termwise product of the sequences. Bennett conjectured that the sequences {\bf x}_n={A-n \choose a} and {\bf y}_n={B-n \choose b} form such a pair for any nonnegative integersA⩾a,B⩾b, and proved this result in the cases min{A, B}⩾a+band −b⩽A−B⩽a. We complete the proof of the conjecture by proving the result under the assumption max{A, B}⩾a+b
Proof of a Conjecture of Frankl and Füredi

AbstractWe give a simple linear algebraic proof of the following conjecture of Frankl and Füredi [7, 9, 13]. (Frankl–Füredi Conjecture) If F is a hypergraph onX={1, 2, 3, …, n} such that1⩽|E∩F|⩽k∀E, F∈F,E≠F,then |\cal F|\le\sum^k_{i=0} {n-1 \choose i}. We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace ofR|F|. Finally, the desired bound on |F| is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray–Chaudhuri–Wilson theorem
Yang–Baxter Type Equations and Posets of Maximal Chains

AbstractThis paper addresses the problem of constructing higher dimensional versions of the Yang–Baxter equation from a purely combinatorial perspective. The usual Yang–Baxter equation may be viewed as the commutativity constraint on the two-dimensional faces of a permutahedron, a polyhedron which is related to the extension poset of a certain arrangement of hyperplanes and whose vertices are in 1–1 correspondence with maximal chains in the Boolean poset Bn. In this paper, similar constructions are performed in one dimension higher, the associated algebraic relations replacing the Yang–Baxter equation being similar to the permutahedron equation. The geometric structure of the poset of maximal chains inSa1×…×Sakis discussed in some detail, and cell types are found to be classified by a poset of “partitions of partitions” in much the same way as those for permutahedra are classified by ordinary partitions
Intersection Statements for Systems of Sets

AbstractA family ofrsets is called aΔ-system if any two sets have the same intersection. Denote byF(n, r) the most number of subsets of ann-element set which do not contain aΔ-system consisting ofrsets. Constructive new lower bounds forF(n, r) are given which improve known probabilistic results, and a new upper bound is given by employing an argument due to Erdős and Szemerédi. Another construction is given which shows that for certainn,F(n, 3)⩾1.551n−2. We also show a relationship between the upper bound forF(n, 3) and the Erdős–Rado conjecture on the largest uniform family of sets not containing aΔ-system
Anti-Hadamard Matrices, Coin Weighing, Threshold Gates, and Indecomposable Hypergraphs

AbstractLetχ1(n) denote the maximum possible absolute value of an entry of the inverse of annbyninvertible matrix with 0,1 entries. It is proved thatχ1(n)=n(1/2+o(1))n. This solves a problem of Graham and Sloane. Letm(n) denote the maximum possible numbermsuch that given a set ofmcoins out of a collection of coins of two unknown distinct weights, one can decide if all the coins have the same weight or not usingnweighings in a regular balance beam. It is shown thatm(n)=n(1/2+o(1))n. This settles a problem of Kozlov and Vũ. LetD(n) denote the maximum possible degree of a regular multi-hypergraph onnvertices that contains no proper regular nonempty subhypergraph. It is shown thatD(n)=n(1/2+o(1))n. This improves estimates of Shapley, van Lint and Pollak. All these results and several related ones are proved by a similar technique whose main ingredient is an extension of a construction of Håstad of threshold gates that require large weights
Generalized Tree Inversions andk-Parking Functions

AbstractKreweras studied a polynomialPn(q) which enumerates (labeled) rooted forests by number of inversions, as well as complements of parking functions by the sum of their terms. Moreover,Pn(1+q) enumerates labeled connected graphs by their number of excess edges. For any positive integerk, there are known notions ofk-parking functions and of (labeled) rootedk-forests, generating the casek=1 studied by Kreweras. We show that the enumeratorP(k)n(q) for complements ofk-parking functions by the sum of their terms is identical to the enumerator ofI(k)n(q) of rootedk-forests by the number of their inversions. In doing so we find recurrence relations satisfied byP(k)n(q) andI(k)n(q), and we introduce the concept of amultirooted k-graphwhose excess edges and roots are enumerated by a polynomial denotedC(k)n(q). We show thatC(k)n(q) satisfies the same recurrence relations as bothP(k)n(1+q) andI(k)n(1+q), proving thatP(k)n(q)=I(k)n(q)
Ramsey Properties of Random Hypergraphs

AbstractLetK(k)(n,p) be the randomk-uniform hypergraph obtained by independent inclusion of each of the[formula]k-tuples with probabilityp. For an arbitraryk-uniform hypergraphGand every integerrwe find the threshold for the property that everyr-coloring of the vertices ofK(k)(n,p) results in a monochromatic copy ofG. In the edge coloring case, which is of our main interest, we find the threshold only fork=3,r=2, andG=K34. In the proof we utilize a recent version of the hypergraph regularity lemma due to Frankl and Rödl
Expansion Properties of Cayley Graphs of the Alternating Groups

AbstractLetCbe a conjugacy class in the alternating groupAn, and let supp(C) be the number of nonfixed digits under the action of a permutation inC. For every 1>δ>0 andn⩾5 there exists a constantc=c(δ)>0 such that if supp(C)⩾δnthen the undirected Cayley graphX(An, C) is acexpander. A family of such Cayley graphs withsupp(C)=o(n)is not a family ofc-expanders. For everyδ>0, ifsupp(C)⩾nthen sets of vertices of order at most(12−δ)(n−(n/supp(C)))!inX(An, C) expand. The proof of the last result combines spectral and representation theory techniques with direct combinatorial arguments
Random Set Partitions: Asymptotics of Subset Counts

AbstractWe study the asymptotics of subset counts for the uniformly random partition of the set [n]. It is known that typically most of the subsets of the random partition are of sizer, withrer=n. Confirming a conjecture formulated by Arratia and Tavaré, we prove that the counts of other subsets are close, in terms of the total variation distance, to the corresponding segments of a sequence {Zj} of independent, Poisson (rj/j!) distributed random variables. DeLaurentis and Pittel had proved that the finite–dimensional distributions of a continuous time process that counts the typical size subsets converge to those of the Brownian Bridge process. Combining the two results allows to prove a functional limit theorem which covers a broad class of the integral functionals. Among illustrations, we prove that the total number of refinements of a random partition is asymptotically lognormal
On Maximal Closed Subsets in Association Schemes

AbstractWe offer a sufficient condition for a closed subset in an association scheme to be maximal. The result generalizes naturally the well-known (group-theoretical) fact that the one-point-stabilizer of a flag transitive automorphism group of a 2-design withλ=1 must be a maximal subgroup. In finite group theory, there exist a lot of (important) sufficient conditions for a subgroup to be maximal. In the theory of association schemes, the condition which we offer here seems to be the first one which guarantees that a closed subset of the set of relations of an association scheme is maximal
On the Size of Set Systems on [n] Not Containing Weak (r,Δ)-Systems

AbstractLetr⩾3 be an integer. A weak (r,Δ)-system is a family ofrsets such that all pairwise intersections among the members have the same cardinality. We show that fornlarge enough, there exists a family F of subsets of [n] such that F does not contain a weak (r,Δ)-system and |F|⩾2(1/3)·n1/5log4/5(r−1). This improves an earlier result of Erdős and Szemerédi (1978,J. Combin. Theory Ser. A24, 308–313; cf. Erdős, On some of my favorite theorems, in “Combinatorics, Paul Erdős Is Eighty,” Vol. 2, Bolyai Society Math. Studies, pp. 97–133, János Bolyai Math. Soc., Budapest, 1990)
A Unifying Construction for Difference Sets

AbstractWe present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v,k,λ,n)=(22d+4(22d+2−1)/3, 22d+1(22d+3+1)/3, 22d+1(22d+1+1)/3, 24d+2) ford⩾0. The construction establishes that a McFarland difference set exists in an abelian group of order 22d+3(22d+1+1)/3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of orderpr. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly forr>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylowp-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups
Thec-2d-Index of Oriented Matroids

AbstractWe obtain an explicit method to compute thecd-index of the lattice of regions of an oriented matroid from theab-index of the corresponding lattice of flats. Since thecd-index of the lattice of regions is a polynomial in the ring Z(c,2d), we call it thec-2d-index. As an application we obtain a zonotopal analogue of a conjecture of Stanley: among all zonotopes the cubical lattice has the smallestc-2d-index coefficient-wise. We give a new combinatorial description for thec-2d-index of the cubical lattice and thecd-index of the Boolean algebra in terms of all the permutations in the symmetric groupSn. Finally, we show that only two-thirds of theα(S)'sof the lattice of flats are needed for thec-2d-index computation
A Complementation Theorem for Perfect Matchings of Graphs Having a Cellular Completion

AbstractA cellular graph is a graph whose edges can be partitioned into 4-cycles (called cells) so that each vertex is contained in at most two cells. We present a “Complementation Theorem” for the number of matchings of certain subgraphs of cellular graphs. This generalizes the main result of M. Ciucu (J. Algebraic Combin.5(1996), 87–103). As applications of the Complementation Theorem we obtain a new proof of Stanley's multivariate version of the Aztec diamond theorem, a weighted generalization of a result of Knuth (J. Algebraic Combin.6(1997), 253–257) concerning spanning trees of Aztec diamond graphs, a combinatorial proof of Yang's enumeration (“Three Enumeration Problems Concerning Aztec Diamonds,” Ph.D. thesis, M.I.T., 1991) of matchings of fortress graphs and direct proofs for certain identities of Jockusch and Propp
Exact Enumeration of 1342-Avoiding Permutations: A Close Link with Labeled Trees and Planar Maps

AbstractSolving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating functionH(x) of all 1342-avoiding permutations of lengthnas well as anexactformula for their numberSn(1342). While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of lengthnequals that of labeled plane trees of a certain type onnvertices recently enumerated by Cori, Jacquard, and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte (Can. J. Math.33(1963), 249–271). Moreover,H(x) turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Noonan and Zeilberger (Adv. Appl. Math.17(1996), 381–407). We also prove thatSn(1342)converges to 8, so in particular, limn→∞(Sn(1342)/Sn(1234))=0
Parameter Augmentation for Basic Hypergeometric Series, II

AbstractIn a previous paper, we explored the idea of parameter augmentation for basic hypergeometric series, which provides a method of provingq-summation and integral formula based special cases obtained by reducing some parameters to zero. In the present paper, we shall mainly deal with parameter augmentation forq-integrals such as the Askey–Wilson integral, the Nassrallah–Rahman integral, theq-integral form of Sears transformation, and Gasper's formula of the extension of the Askey–Roy integral. The parameter augmentation is realized by another operator, which leads to considerable simplications of some well knownq-summation and transformation formulas. A brief treatment of the Rogers–Szegö polynomials is also given
Asymptotic Enumeration of Convex Polygons

AbstractA polygon is an elementary (self-avoiding) cycle in the hypercubic lattice Zdtaking at least one step in every dimension. A polygon on Zdis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonspn,dof length 2nwithd(n)→∞ is asymptoticallypn,d∼exp−2(2n−d)2n−1(2n−1)!(2πb(r))−1/2r−2n+dsinhdr,wherer=r(n,d) is the unique solution ofrcothr=2n/d−1andb(r)=d(rcothr−r2/sinh2r). The convergence is uniform over alld⩾ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d−1)2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon
Results on Bent Functions

AbstractIn this paper, we present three results on bent functions: a construction, a restriction, and a characterization. Starting with a single bent function, in a simple but very effective way, the construction produces a large number of new bent functions in the same number of variables. The restriction imposes new conditions on the directional derivatives of bent functions. Certain non-existence results that were previously obtained through computer search follow easily from these conditions. The characterization describes bent functions as certain solutions of a system of quadratic equations. Interesting new properties of bent functions are obtained using the characterization