pinecone/core-2020-05-10-deduplication huggingface dataset:
A Proof of then! Conjecture for Generalized Hooks

AbstractIn [4], Garsia and Haiman [Electronic J. of Combinatorics3, No. 2 (1996)] pose a conjecture central to their study of the Macdonald polynomialsHμ(x; q, t). For eachμ⊢none defines a certain determinantΔμ(Xn, Yn) in two sets of variables. Then! conjecture asserts that the vector space given by the linear span of derivatives ofΔμ, written L[∂px∂qyΔμ(Xn, Yn)], has dimensionn!. The conjecture reduces to a well-known result about the Vandermonde determinant whenμ=(1n) orμ=(n) (see [1], for example). Garsia and Haiman (see [Proc. Natl. Acad. Sci.90(1993), 3607–3610]) have demonstrated the conjecture for two-rowed shapesμ=(a, b), two-columned shapesμ=(2a, 1b) and hook shapesμ=(a, 1b). In this paper, we give an overview of the methods used by Reiner in his thesis to prove then! conjecture for generalized hooks, that is, forμ=(a, 2, 1b)
Long Symmetric Chains in the Boolean Lattice

AbstractLet 2[n]be the poset of all subsets of a set with n elements ordered by inclusion. A long chain in this poset is a chain ofn−1 subsets starting with a subset with one element and ending with a subset withn−1 elements. In this paper we prove: Given any collection of at mostn−2 skipless chains in 2[n], there exists at least one (but sometimes not more than one) long chain disjoint from the chains in the collection. Furthermore, fork⩾3, given a collection ofn−kskipless chains in 2[n], there are at leastkpairwise disjoint long chains which are also disjoint from the given chains
Noncommutative Cyclic Characters of Symmetric Groups

AbstractWe define noncommutative analogues of the characters of the symmetric group which are induced by transitive cyclic subgroups (cyclic characters). We investigate their properties by means of the formalism of noncommutative symmetric functions. The main result is a multiplication formula whose commutative projection gives a combinatorial formula for the resolution of the Kronecker product of two cyclic representations of the symmetric group. This formula can be interpreted as a multiplicative property of the major index of permutations
Small Complete Caps in Spaces of Even Characteristic

AbstractIn 1959, Segre constructed a complete (3q+2)-cap inPG(3, q),qeven. This showed that the size of the smallest completek-cap inPG(3, q),qeven, is almost equal to the trivial lower bound which is of order[formula]. Generalizing the construction of Segre, complete (qn+3(qn−1+…+q)+2)-caps inPG(2n, q),qeven,q⩾4, and complete (3(qn+…+q)+2)-caps inPG(2n+1, q),qeven,q⩾4, are constructed. This shows that in all spacesPG(2n+1, q),qeven, the size of the smallest completek-cap is almost equal to the trivial lower bound which is of order[formula]
Random Covering Designs

AbstractAt−(n, k, λ) covering design (n⩾k>t⩾2) consists of a collection ofk-element subsets (blocks) of ann-element set X such that eacht-element subset of X occurs in at leastλblocks. Letλ=1 andk⩽2t−1. Consider a randomly selected collection B of blocks; |B|=φ(n). We use the correlation inequalities of Janson to show that B exhibits a rather sharp threshold behaviour, in the sense that the probability that it constitutes at−(n, k, 1) covering design is, asymptotically, zero or one—according asφ(n)={(nt)/(kt)}(log(nt)−ω(n)) orφ(n)={(nt)/(kt)}(log(nt)+ω(n)), whereω(n)→∞ is arbitrary. We then use the Stein–Chen method of Poisson approximation to show that the restrictive conditionk⩽2t−1 in the above result can be dispensed with. More generaly, we prove that if each block is independently “selected” with a certain probabilityp, the distribution of the numberWof uncoveredtsets can be approximated by that of a Poisson random variable provided thatE|B|⩾{(nt)/(kt)}[(t−1)logn+loglogn+an], wherean→∞ at an arbitrarily slow rate
Neighborly 2-Manifolds with 12 Vertices

AbstractWe provide a complete list of 59 orientable neighborly 2-manifolds with 12 vertices of genus 6, and we study their possible flat embeddings in Euclidean 3-space. Whereas the question of embeddability remains open in its general form, we obtain several properties of the embedding (polyhedral realization) under the assumption that it does exist: 1.The order of the geometrical automorphism group of any polyhedral realization would not exceed 2. 2.The polyhedral realization would not be obtainable via the Schlegel diagram of any 4-polytope; moreover, none of our orientable neighborly 2-manifolds with 12 vertices can be found within of the 2-skeleton of any 4-polytope. 3.The polyhedral realization would not allow a tetrahedral subdivision without inserting new vertices. By using a weaker version of the manifold property, we obtain neighborly polyhedra with 2nvertices for everyn⩾3
Raney Paths and a Combinatorial Relationship between Rooted Nonseparable Planar Maps and Two-Stack-Sortable Permutations

AbstractAn encoding of the set of two-stack-sortable permutations (TSS) in terms of lattice paths and ordered lists of strings is obtained. These lattice paths are called Raney paths. The encoding yields combinatorial decompositions for two complementary subsets of TSS, which are the analogues of previously known decompositions for the set of nonseparable rooted planar maps (NS). This provides a combinatorial relationship between TSS and NS, and, hence, a bijection is determined between these sets that is different, simpler, and more refined than the previously known bijection
A Finite Generalized Hexagon Admitting a Group Acting Transitively on Ordered Heptagons is Classical

AbstractLetΓbe a thick finite generalized hexagon and letGbe a group of automorphisms ofΓ. IfGacts transitively on the set of non-degenerate ordered heptagons, thenΓis one of the Moufang hexagonsH(q) or3H(q) associated to the Chevalley groupsG2(q) or3D4(q) respectively, or their duals; andGcontains the corresponding Chevalley group. Moreover, we show that no thick generalized octagon admitting a group acting transitively on the set of ordered nonagons (enneagons) can exist. This completes the determination of all finite thick generalizedn-gons,n≥3, with a group acting transitively on the set of ordered (n+1)-gons with elementary methods. Because we do not use the classification of the finite simple groups, from which these results also follow
Asymptotically Optimal Covering Designs

AbstractA (v,k,t)covering design, orcovering, is a family ofk-subsets, calledblocks, chosen from av-set, such that eacht-subset is contained in at least one of the blocks. The number of blocks is the covering'ssize, and the minimum size of such a covering is denoted byC(v,k,t). It is easy to see that a covering must contain at least (tv)/(tk) blocks, and in 1985 Rödl [5] proved a long-standing conjecture of Erdos and Hanani [3] that for fixedkandt, coverings of size (tv)/(tk)(1+o(1)) exist (asv→∞). An earlier paper by the first three authors [4] gave new methods for constructing good coverings, and gave tables of upper bounds onC(v,k,t) for smallv,k, andt. The present paper shows that two of those constructions are asymptotically optimal: For fixedkandt, the size of the coverings constructed matches Rödl's bound. The paper also makes theo(1) error bound explicit, and gives some evidence for a much stronger bound
Four Symmetry Classes of Plane Partitions under One Roof

AbstractIn a previous paper, the author applied the permanent-determinant method of Kasteleyn and its non-bipartite generalization, the Hafnian–Pfaffian method, to obtain a determinant or a Pfaffian that enumerates each of the ten symmetry classes of plane partitions. After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane partitions, and the intersection of the last two types) in the representation theory of sl(2, C). The result is a unified proof of the four enumerations
A Combinatorial Laplacian with Vertex Weights

AbstractOne of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 17, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L forGis a matrix with rows and columns indexed by the vertex setVofG, and the (u,v)-entry of L, foru,vinG,u≠v, is associated with the edge-weight of the edge (u,v). It is not so obvious to consider Laplacians with vertex weights (except for using some symmetric combinations of the vertex weights to define edge-weights). In this note, we consider a vertex weighted Laplacian which is motivated by a problem arising in the study of algebro-geometric aspects of the Bethe Ansatz [18]. The usual Laplacian can be regarded as a special case with all vertex-weights equal. We will generalize the matrix-tree theorem to matrix-tree theorems of counting “rooted” directed spanning trees. In addition, the characteristic polynomial of the vertex-weighted Laplacian has coefficients with similar interpretations. We also consider subgraphs with non-trial boundary. We will shown that the Laplacian with Dirichlet boundary condition has its determinant equal to the number of rooted spanning forests. The usual matrix-tree theorem is a special case with the boundary consisting of one single vertex
Pfaffians and Determinants for SchurQ-Functions

AbstractSchurQ-functions were originally introduced by Schur in relation to projective representations of the symmetric group, and they can be defined combinatorially in terms of shifted tableaux. In this paper we describe planar decompositions of shifted tableaux into strips and use the shapes of these strips to generate pfaffians and determinants that are equal to SchurQ-functions. As special cases we obtain the classical pfaffian associated with SchurQ-functions, a pfaffian for skewQ-functions due to Jozefiak and Pragacz, and a determinantal expression of Okada
Maximal Subgroups of Symmetric Groups

AbstractWe show thatSnhas at mostn6/11+o(1)conjugacy classes of primitive maximal subgroups. This improves annclog3nbound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups ofSnis of the form (12+o(1))n. It also follows that, for largen,Snhas less thann ! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at mostn17/11+o(1)maximal subgroups of indexn
Tableau Switching: Algorithms and Applications

AbstractWe define and characterizeswitching, an operation that takes two tableaux sharing a common border and “moves them through each other” giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood–Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schützenberger's evacuation procedure to switching and in the process obtain further results concerning evacuation. We definereversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood–Richardson coefficients
Permutation Polynomials, de Bruijn Sequences, and Linear Complexity

AbstractThe paper establishes a connection between the theory of permutation polynomials and the question of whether a de Bruijn sequence over a general finite field of a given linear complexity exists. The connection is used both to construct span 1 de Bruijn sequences (permutations) of a range of linear complexities and to prove non-existence results for arbitrary spans. Upper and lower bounds for the linear complexity of a de Bruijn sequence of spannover a finite field are established. Constructions are given to show that the upper bound is always tight, and that the lower bound is also tight in many cases
Binomial andQ-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and TheirQ-Analogues

AbstractThe Kneser graphK(n, k) has as vertices all thek-subsets of a fixedn-set and has as edges the pairs {A, B} of vertices such thatAandBare disjoint. It is known that these graphs are Hamiltonian if[formula]forn⩾2k+1. We determine asymptotically for fixedkthe minimum valuen=e(k) for which this inequality holds. In addition we give an asymptotic formula for the solution ofkΓ(n)Γ(n−2k+1)=Γ2(n−k+1) forn⩾2k+1, ask→∞, whennandkare not restricted to take integer values. We also show that for all prime powersqandn⩾2k,k⩾1, theq-analoguesKq(n, k) are Hamiltonian by consideration of the analogous inequality forq-binomial coefficients
On Increasing Subsequences of Random Permutations

AbstractLetLnbe the length of a longest increasing subsequence in a random permutation of {1, …, n}. It is known that the expected value ofLnis asymptotically equal to[formula]asngets large. This note derives upper bound on the probability that[formula]exceeds certain quantities. In particular, we prove that[formula]is at most ordern1/6with high probability. Our main result is an isoperimetric upper bound of the probability that[formula]exceedθn1/6, which suggests that the varianceV[Ln] might ben1/3. We also find an explicit lower bound of the function[formula], defined by D. Aldous and P. Diaconis
An Equivalence between (T, M, S)-Nets and Strongly Orthogonal Hypercubes

AbstractIt is well known that (t, m, s)-nets are useful in numerical analysis. While many of the best constructions of such nets arise from number theoretic or algebraic constructions, we will show in this paper that the existence of a (t, t+k, s)-net in basebis equivalent to the existence of a set of s strongly orthogonal hypercubes of dimensiont+k, orderband strengthk. Fork>2 such generalized orthogonal hypercubes provide new combinatorial structures that may be of interest in various other combinatorial settings
Constructions of Partial Difference Sets and Relative Difference Sets Using Galois Rings II

AbstractIn a previous paper, [Des., Codes and Cryptogr.8(1996), 215–227]; we used Galois rings to construct partial difference sets, relative difference sets and a difference set. In the present paper, we first generalize and improve the construction of partial difference sets in [Des., Codes and Cryptogr.8(1996), 215–227]; also we obtain a family of relative difference sets from these partial difference sets. Second, we construct a class of relative difference sets in (Z4)2m+1⊕(Z4)r⊕(Z2⊕Z2)s,r+s=m, r, s⩾0 with respect to a subgroup (Z2)2m+1. These constructions make use of character sums from Galois rings, and relate relative difference sets to Hadamard difference sets
On the Existence of de Bruijn Tori with Two by Two Windows

AbstractNecessary and sufficient conditions for the existence of de Bruijn Tori (or Perfect Maps) with two by two windows over any alphabet are given. This is the first two-dimensional window size for which the existence question has been completely answered for every alphabet. The techniques used to construct these arrays utilise existing results on Perfect Factors and Perfect Multi-Factors in one and two dimensions and involve new results on Perfect Factors with ‘puncturing capabilities’. Finally, the existence question for two-dimensional Perfect Factors is considered and is settled for two by two windows and alphabets of prime-power size
Combinatorics of theq-Basis of Symmetric Functions

AbstractA basis of symmetric functions, which we denote byqλ(X; q, t), was introduced in the work of Ram and King and Wybourne in order to describe the irreducible characters of the Hecke algebras of type A. In this work we give combinatorial descriptions of the expansions of the functionsqλ(X; q, t) in terms of the classical bases of symmetric functions and apply these results in determining the determinant of the character table of the Iwahori–Hecke algebras and in giving a generating function for certain permutation statistics
Remarks on a Zero-Sum Theorem

AbstractRecently the following theorem in combinatorial group theory has been proved: LetGbe a finite abelian group and letAbe a sequence of members ofGsuch that |A|⩾|G|+D(G)−1, whereD(G) is the Davenport constant ofG. ThenAcontains a subsequenceBsuch that |B|=|G| and ∑b∈Bb=0. We shall present a generalization of this theorem which contains information on the extremal cases and in particular allows us to deduce a short proof of the extremal cases in the Erdo&#x030B;s–Ginzburg–Ziv theorem. We also present, using the above-mentioned theorem, a proof that ifGhas rankkthen |A|⩾|G|(1+(k+1)/2k)−1 suffices to ensure a zero-sum subsequence on |G| terms
Lattice Path Proofs for Determinantal Formulas for Symplectic and Orthogonal Characters

AbstractWe give bijective proofs for Jacobi–Trudi-type and Giambelli-type identities for symplectic and orthogonal characters. These proofs base on interpreting King and El-Sharkaway's symplectic tableaux, Proctor's odd and intermediate symplectic tableaux, Proctor's and King and Welsh's orthogonal tableaux, and Sundaram's odd orthogonal tableaux in terms of certain families of nonintersecting lattice paths. This work is intended to be the counterpart of the Gessel–Viennot proof of the Jacobi–Trudi identities for Schur functions for the case of symplectic and orthogonal characters
Enumeration of Perfect Matchings in Graphs with Reflective Symmetry

AbstractA plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch's squarishness theorem. As applications of our result, we enumerate the perfect matchings of several families of graphs and we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions (namely, transposed complementary and cyclically symmetric, transposed complementary) contained in a given box. Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees
An Entropy Proof of Bregman's Theorem

AbstractLetA=(ai, j) be ann×n0-1 matrix. LetSbe the set of permutationsσof [n] such thatai, σ(i)=1 fori=1, 2, …, n. Then, the permanent ofAis perm(A)=def|S|. Theorem(Brégman,Soviet Math. Dokl.14(1973), 945–949). If the number of 1's in rowiofAisri, then[formula]A short proof of this theorem was given by Schrijver (J. Combin. Theory Ser. A25(1978), 80–83); Alon and Spencer (“The Probabilistic Method,” Wiley/Interscience, New York, 1992), obtained a similar proof by analyzing a randomized procedure for estimating the permanent. In this note we present a proof based on entropy
On the Degree, Size, and Chromatic Index of a Uniform Hypergraph

AbstractLet H be ak-uniform hypergraph in which no two edges share more thantcommon vertices, and letDdenote the maximum degree of a vertex of H. We conjecture that for everyε>0, ifDis sufficiently large as a function oft, k, andε, then the chromatic index of H is at most (t−1+1/t+ε)D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form: If H is an intersectingk-uniform hypergraph in which no two edges share more thantcommon vertices andDis the maximum degree of a vertex of H, whereDis sufficiently large as a function ofk, then H has at most (t−1+1/t)Dedges
Schützenberger's Jeu de Taquin and Plane Partitions

AbstractWe modify Schützenberger's “jeu de taquin” and Knuth's generalization DELETE of the Robinson–Schensted correspondence to apply to unrestricted rather than just column-strict plane partitions. The “jeu de taquin,” DELETE, their modifications, and the Hillman–Grassl mapping are essentially equivalent. We extend the combinatorial methods of Bender and Knuth to give an extension of an elegant, unpublished result of Stanley. Our main result is equivalent to the evaluation of the generating function for column-strict plane partitions of fixed shape with parts less than or equal tom. We prove MacMahon's “box” theorem and give a generating function for plane partitions with parts less than or equal tomand the parts below rowrform a column-strict plane partition with at mostccolumns
Constructions of Hadamard Difference Sets

AbstractUsing a spread ofPG(3, p) and certain projective two-weight codes, we give a general construction of Hadamard difference sets in groupsH×(Zp)4, whereHis either the Klein 4-group or the cyclic group of order 4, andpis an odd prime. In the casep≡3 (mod 4), we use an ovoidal fibration ofPG(3, p) to construct Hadamard difference sets, this construction includes Xia's construction of Hadamard difference sets as a special case. In the casep≡1 (mod 4), we construct new reversible Hadamard difference sets by explicitly constructing the two-weight codes needed in our general construction method. Using a well-known composition theorem, we conclude that there exist Hadamard difference sets with parameters (4m2, 2m2−m, m2−m), wherem=2a3b52c1132c2172c3p21p22…p2twitha, b, c1, c2, c3positive integers and where eachpjis a prime congruent to 3 modulo 4, 1⩽j⩽t
All Z4Codes of Type II and Length 16 Are Known

AbstractWe show that there are 133 inequivalent Type II codes over Z4of length 16. We give the number of each type 4i·2j, where 2i+j=16, a generator matrix for each code, the order of its automorphism group, and its minimum Lee weight. A (partial) symmetrized weight enumerator of each code is given. The highest minimum Lee weight amongst these codes is 8, and there are 5 inequivalent codes attaining this highest Lee weight. A new computer algorithm to determine the automorphism group of a Z4code was use
Independent Finite Sums forKm-Free Graphs

AbstractRecently, in conversation with Erdős, Hajnal asked whether or not for any triangle-free graphGon the vertex set N, there always exists a sequence ⦠xn⦔∞n=1so that wheneverFandHare distinct finite nonempty subsets of N, {∑n∈Fxn, ∑n∈Hxn} is not an edge ofG(that is,FS(⦠xn⦔∞n=1) is an independent set). We answer this question in the negative. We also show that if one replaces the assumption thatGis triangle-free by the assertion that for somem,Gcontains no complete bipartite graphKm, m, then the conclusion does hold. If for somem⩾3,Gcontains noKm, we show there exists a sequence ⦠xn⦔∞n=1so that wheneverFandHare disjoint finite nonempty subsets of N, {∑n∈Fxn, ∑n∈Hxn} is not an edge ofG. Both of the affirmative results are in fact valid for a graphGon an arbitrary cancellative semigroup (S, +)
Applications of Coding Theory to the Construction of Modular Lattices

AbstractWe study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different nonzero values is defined over these rings; using invariant theory, we give a basis for the space of invariants to which belongs the three variables weight enumerator of a self-dual code. A general bound for the weight of such codes is derived. We construct a number of extremal self-dual codes, which are the codes reaching this bound, and derive some extremal lattices of levell=2, 3, 7 and minimum 4, 6, 8