pinecone/core-2020-05-10-deduplication huggingface dataset:
Additive and Multiplicative Ramsey Theory in the Reals and the Rationals

AbstractLet a finite partition F of the real interval (0,1) be given. We show that if every member of F is measurable or if every member of F is a Baire set, then one member of F must contain a sequence with all of its finite sums and products (and, in the measurable case, all of its infinite sums as well). These results are obtained by using the algebraic structure of the Stone–Čech compactification of the real numbers with the discrete topology. They are also obtained by elementary methods. In each case we in fact get significant strengthenings of the above stated results (with different strengthenings obtained by the algebraic and elementary methods). Some related (although weaker) results are established for arbitrary partitions of the rationals and the dyadic rationals, and a counterexample is given to show that even weak versions of the combined additive and multiplicative results do not hold in the dyadic rationals
The Return of the Baer Subplane

AbstractWe discuss the notion of a tangency set in a projective plane, generalising the well-studied idea of a minimal blocking set. Tangency sets have recently been used in connection with the coding theory related to algebraic curves over finite fields, and they are closely related to the strong representative systems introduced by T. Illés, T. Szonyi, and F. Wettl (1991,Mitt. Math. Sem. Giessen201, 97–107). Here we give bounds on the possible sizes of tangency sets, and structural results are obtained in the extremal cases
Monotone Subsequences in Any Dimension

AbstractWe exhibit sequences ofnpoints inddimensions with no long monotone subsequences, by which we mean when projected in a general direction, our sequence has no monotone subsequences of lengthn+dor more. Previous work proved that this function ofnwould lie betweennand 2n; this paper establishes that the coefficient ofnis one. This resolves the question of how the Erdo&#x030B;s–Szekeres result that a (one-dimensional) sequence has monotone subsequences of at mostngeneralizes to higher dimensions
Isotypic Decompositions of Lattice Determinants

AbstractTheq,t-Macdonald polynomials are conjectured by Garsia and Haiman to have a representation theoretic interpretation in terms of theSn-moduleMμspanned by the derivatives of a certain polynomialΔμ(x1,x2,…,xn;y1,y2,…,yn). The diagonal action of a permutationσ∈Snon a polynomialP=P(x1,x2,…,xn;y1 ,y2,…,yn) is defined by settingσP=P(xσ1,xσ2< F,…,xσn;yσ1,yσ2,…,yσn). Since the polynomialΔμalternates under the diagonal action,MμisSn-invariant. We analyze here the diagonal action ofSnonMμand show that its decomposition into irreducibles is equivalent to a certain isotypic expansion for the translateΔμ(x1+x′1,x2+x ′2,…,xn+x′n;y1+y′1,y2+y′2,…,yn +y′n) of the polynomialΔμ
Multilinear Polynomials and a Conjecture of Frankl and Füredi

AbstractFrankl and Füredi conjectured that given a family F of subsets of [n] such that 1⩽|E∩F|⩽kfor all distinctEandFin F, we must have |F|⩽∑ki=0(n−1i), (1981, P. Frankl and Z. Füredi,Collog. Math. Soc. János Bolyai37, 305–320). The first proof of this result was given by G. V. Ramanan in (1997,J. Combin. Ser. A79, 53–67). In this note, we present a proof which is a modification of an approach to this problem by H. Snevily (1994,J. Combin. Ser. A.68, 232–238) and like Snevily's, is based on the technique of N. Alon, L. Babai, and H. Suzuki (1991,J. Combin. Theory Ser. A58, 165–180)
Coalescent Random Forests

AbstractVarious enumerations of labeled trees and forests, including Cayley's formulann−2for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the followingcoalescent constructionof a sequence of random forests (Rn,Rn−1,…,R1) such thatRkhas uniform distribution over the set of all forests ofkrooted trees labeled by [n]. LetRnbe the trivial forest withnroot vertices and no edges. Forn⩾k⩾2, given thatRn,…,Rkhave been defined so thatRkis a rooted forest ofktrees, defineRk−1by addition toRkof a single edge picked uniformly at random from the set ofn(k−1) edges which when added toRkyield a rooted forest ofk−1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, theadditive coalescentin which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitudesxandyrunning a risk at ratex+yof a coalescent collision resulting in a mass of magnitudex+y. The transition semigroup of the additive coalescent is shown to involve probability distributions associated with a multinomial expansion over rooted forests
Non-averaging Subsets and Non-vanishing Transversals

AbstractIt is shown that every set ofnintegers contains a subset of sizeΩ(n1/6) in which no element is the average of two or more others. This improves a result of Abbott. It is also proved that for everyε>0 and everym>m(ε) the following holds. IfA1,…,Amaremsubsets of cardinality at leastm1+εeach, then there area1∈A1,…,am∈Amso that the sum of every nonempty subset of the set {a1,…,am} is nonzero. This is nearly tight. The proofs of both theorems are similar and combine simple probabilistic methods with combinatorial and number theoretic tools
An Injection for the Ehrenpreis Rogers–Ramanujan Problem

AbstractGarsia and Milne used their elegant involution principle to give a bijective proof of the first Rogers–Ramanujan identity. We give an injection as called for by Ehrenpreis of the partitions ofninto parts of the forms 5m+2 and 5m+3 into the partitions ofninto parts of the forms 5m+1 and 5m+4. As observed by Ehrenpreis, Andrews and Baxter, this gives a potential start to a Rogers–Ramanujan bijection and a new partition identity involving partitions into parts ⩾3 with difference between parts at least 2. Our potential bijection does not agree with the Garsia–Milne bijection
Resistance Bounds for First-Passage Percolation and Maximum Flow

AbstractSuppose that each edgeeof a network is assigned a random exponential passage time with meanre. Then the expected first-passage time between two vertices is at least the effective resistance between them for the edge resistances re. Similarly, suppose each edge is assigned a random exponential edge capacity with meance. Then the expected maximum flow between two vertices is at least the effective conductance between them for the edge conductances ce. These inequalities are dual to each other for planar graphs and the second is tight up to a factor of 2 for trees; this has implications for a herd of gnus crossing a river delta
Diagram Rules for the Generation of Schubert Polynomials

AbstractWe prove an elegant combinatorial rule for the generation of Schubert polynomials based on box diagrams, which was conjectured by A. Kohnert. The main tools for the proof are (1) a recursive structure of Schubert polynomials and (2) a partial order on the set of box diagrams. As a byproduct we obtain (combinatorial) proofs for two other rules for the generation of Schubert polynomials based on box diagrams: (1) the more complicated rule of N. Bergeron, and (2) the rule of P. Magyar, which we show to be a simplified Bergeron rule. The well-known fact that the Schubert polynomials associated to Grassmannian permutations are in fact Schur polynomials is derived from Kohnert's rule
A Refinement of the Lecture Hall Theorem

AbstractForn⩾1, let Lnbe the set of lecture hall partitions of lengthn, that is, the set ofn-tuples of integersλ=(λ1,…,λn) satisfying 0⩽λ1/1⩽λ2/2⩽…⩽λn/n. Let ⌈λ⌉ be the partition (⌈λ1/1⌉,…,⌈λn/n⌉), and leto(⌈λ⌉) denote the number of its odd parts. We show that the identity∑λ∈Lnq|λ|u|⌈λ⌉|vo(⌈λ⌉)=(1+uvq)(1+uvq2)…(1+uvqn)(1−u2qn+1)(1−u2qn+2)…(1−u2q2n)is equivalent to a refinement of Bott's formula for the affine Coxeter groupCn, obtained by I. G. Macdonald (Math. Ann.199(1972), 161–174) and V. Reiner (Electron. J. Combin.2(1995), R25). The caseu=v=1 of the above identity, called the lecture hall theorem, was proved by us in (Ramanujan J.1(1997), 101–111), and then by Andrews (“Mathematical Essays in Honor of G.-C. Rota,” pp. 1–22, Birkhäuser, Cambridge, MA, 1998). In the present paper, we give two direct proofs of the above identity. The first one is rather short, but requires a bit ofq-calculus; the second one is the first truly bijective proof ever found in the domain of lecture hall partitions. Although we describe our bijection in completely combinatorial terms, it finds its origin in the algebraic context of Coxeter groups. Both proofs are completely independent of all earlier proofs of the Lecture Hall Theorem
On Singular 1-Rotational Steiner 2-Designs

AbstractA Steiner 2-design is 1-rotational over a groupGif it admitsGas an automorphism group fixing one point and acting regularly on the remainder. 1-rotational Steiner 2-designs have come into fashion since 1981, when Phelps and Rosa (Discrete Math.33(1981), 57–66) studied Steiner triple systems that are 1-rotational over the cyclic group. While all 1-rotational Steiner 2-designs con- structed in the past have exactly oneshort block-orbit, in this paper we also consider 1-rotational Steiner 2-designs not having this property. We call themsingularand we show that they are quite rare. In particular, we enumerate all the abelian 1-rotational 2-(49, 4, 1) designs
On the Random Young Diagrams and Their Cores

AbstractAnr-core of a Young diagramλis a residual subdiagram obtained after consecutive removals of the feasibler-long border strips, “rim hooks.” The removal process on the diagramλand the resultingr-core are the essential elements in the Murnaghan–Nakayama formula forχλ, the character of the associated irreducible representation ofSn(n=|λ|), on the conjugacy class {r[n/r]} (n≡0modr). A complete characterization ofr-cores is given, which extends a well known result forr=2. Under an assumption that the partitionλis chosen uniformly at random out of all partitions ofn, it is shown that typically ther-core size is of ordern1/2, while the height and the width are of ordern1/4. Fornchosen uniformly at random between 1 andN, the core boundary scaled byN1/4is proved to converge, in distribution, to a random concave curve which consists ofr−1 line segments
Most Latin Squares Have Many Subsquares

AbstractAk×nLatin rectangle is ak×nmatrix of entries from {1, 2, …, n} such that no symbol occurs twice in any row or column. An intercalate is a 2×2 Latin sub-rectangle. LetN(R) be the number of intercalates inR, a randomly chosenk×nLatin rectangle. We obtain a number of results about the distribution ofN(R) including its asymptotic expectation and a bound on the probability thatN(R)=0. Forε>0 we prove most Latin squares of ordernhaveN(R)⩾n3/2−ε. We also provide data from a computer enumeration of Latin rectangles for smallk, n
Recursion and Explicit Formulas for Particular N-Variable Knop–Sahi and Macdonald Polynomials

AbstractKnop and Sahi simultaneously introduced a family of non-homogeneous, non-symmetric polynomials, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main results of this paper. In particular, we give a complete solution for the case G(k, a, …, a)(x; q, t). A remarkable by-product of our proofs is the discovery that these polynomials satisfy a recursion on the number of variables
Generalized Quadrangles of Order (s, s2), III

AbstractLet S=(P, B, I) be a generalized quadrangle of order (q, q2), q>1, and assume that S satisfies Property (G) at the flag (x, L). If q is odd then S is the dual of a flock generalized quadrangle. This solves (a stronger version of ) a ten-year-old conjecture. We emphasize that this is a powerful theorem as Property (G) is a simple combinatorial property, while a flock generalized quadrangle is concretely described using finite fields and groups. As in several previous theorems it was assumed that the dual of the generalized quadrangle arises from a flock, this can now be replaced, in the odd case, by having Property (G) at some flag. Finally we describe a pure geometrical construction of a generalized quadrangle arising from a flock; until now there was only the construction by Knarr which only worked in the odd case
Some Graded Representations of the Complex Reflection Groups

AbstractLet C[X, Y] denote the ring of polynomials with complex coefficients in the variables X={x1, …, xn} and Y={y1, …, yn}, let Sn denote the symmetric group of order n!, let Cm denote the cyclic group Cm={e2πij/m:0⩽j⩽m−1} of order m, let Hk denote the subgroup of order k of Cm, and let Gn, m=Cm≀Sn (the wreath product of Cm with Sn). Each element ϕ of Gn, m may be represented as a generalized permutation ϕ=[ε(1)σ1, …, ε(n)σn] where ε(j)=e2πihj/m where 0⩽hj⩽m−1 and σ=[σ1, …, σn]∈Sn. Let Gn, m, k={ϕ∈Gn, m:∏ni=1ε(i)∈Hk}. In this paper, alternants Am, k, ν are defined in the variables X and Y and where n, m, and k are integers and ν is a partition. Setting In, m, k, ν(X, Y) to be the ideal In, m, k, ν(X, Y)={P∈C[X, Y]:P(∂x1, …, ∂xn, ∂y1, …, ∂yn)Am, k, ν=0}, where ∂xi (resp. ∂yj) is the partial differential operator with respect to xi (resp. yj), the action of Gn, m, k on the quotient ring Cn, m, k, ν=C[X, Y]/In, m, k, ν(X, Y) is isomorphic to the regular representation of Gn, m, k when ν=(1p), some ν=(p) (and k divides m) or when ν is a hook-shape and k=1. Bases are constructed for these quotient rings that exhibit the decomposition of the regular representation into irreducibles. It should be noted that the alternants Am, k, ν are generalizations of the alternants defined by A. Garsia and M. Haiman (1996, Electron. J. Combin.3, No. 2). Thus the graded characters of Cn, m, k, ν give generalizations of the q, t-Kostka coefficients
Isomorphisms of Finite Cayley Digraphs of Bounded Valency, II

AbstractFor a finite group G and a subset S of G which does not contain the identity of G, denote by Cay(G, S) the Cayley digraph of G with respect to S. An automorphism σ of the group G induces a graph isomorphism from Cay(G, S) to Cay(G, Sσ). In this paper, we investigate groups G and Cayley digraphs Cay(G, S) of G for which the following condition holds: for any T⊂G, Cay(G, S)≅Cay(G, T) if and only if Sσ=T for some σ∈Aut(G). For a positive integer m, a group G is called an m-DCI-group if the condition holds for all Cayley digraphs of valency at most m; while G is called a connected m-DCI-group if it holds for all connected digraphs of valency at most m. This paper contributes towards a complete classification of finite m-DCI-groups for m⩾2. It was previously proved by C. H. Li et al. (1998, J. Combin. Theory Ser. B74, 164–183) that finite m-DCI-groups for m⩾2 belong to an explicitly determined list DCI(m) of groups. However, it is still an open problem to determine which members of DCI(m) are really m-DCI-groups. We reduce this problem to the problem of determining whether all subgroups of groups in DCI(m) are connected m-DCI-groups. Then we give a complete classification of finite 2-DCI-groups
A Solution to the Generalized Cevian Problem Using Forest Polynomials

AbstractIn this paper, we examine the effect of dissecting an n-dimensional simplex using cevians (cross-sections passing through n−1 of the vertices of the simplex). We describe a formula for the number of pieces the simplex is dissected into using a polynomial involving only the number of each type of cevian. The polynomial in question involves terms involving the edges of the simplex, but discarding those terms involving cycles of the underlying graph. Thus, we call such a polynomial a “forest polynomial.
Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs

AbstractThree classes of finite structures are related by extremal properties: complete d-partite d-uniform hypergraphs, d-dimensional affine cubes of integers, and families of 2d sets forming a d-dimensional Boolean algebra. We review extremal results for each of these classes and derive new ones for Boolean algebras and hypergraphs, several obtained by employing relationships between the three classes. Related partition or coloring problems are also studied for Boolean algebras. Density results are given for Boolean algebras of sets all of whose atoms are the same size
On Finite Groups with the Cayley Isomorphism Property, II

AbstractFor a positive integer m, a group G is said to have the m-DCIproperty if, for any Cayley digraphs Cay(G, S) and Cay(G, T) of G of valency m (that is, |S|=|T|=m), Cay(G, S)≅Cay(G, T) if and only if Sσ=T for some σ∈Aut(G). This paper is one of a series of papers towards characterizing finite groups with the m-DCI property. It is shown that, for infinitely many values of m, there exist Frobenius groups with the m-DCI property but not with the k-DCI property for any k<m. Further, it is conjectured that for relative small values of m, these groups and an explicit list of groups given by C. H. Li, C. E. Praeger, and M. Y. Xu (1998, J. Combin. Theory Ser. B73, 164–183) contain all finite groups with the m-DCI property. This conjecture is verified for the case m⩽4
Sign Types Associated to Posets

AbstractWe start with a combinatorial definition of I-sign types which are a generalization of the sign types indexed by the root system of type Al (I⊂N finite). Then we study the set DIp of I-sign types associated to the partial orders on I. We establish a 1–1 correspondence between D[n]p and a certain set of polyhedral cones in a euclidean space by which we get a geometric distinction of the sign types in D[n]p from the other [n]-sign types. We give a graph-theoretical criterion for an Sn-orbit O of D[n]p to contain a dast and show that O contains at most one dast. Finally, we show the admirability of a poset associated to a dast
Another Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula

AbstractAnother bijective proof of Stanley's hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author's previous bijective proof (1998, Discrete Math. Theoret. Comput. Sci.3, 011–032) and the Novelli–Pak–Stoyanovskii bijection (J. C. Novelli et al., 1997, Discrete Math. Theoret. Comput. Sci.1, 53–67) for the hook formula for standard Young tableaux of a given shape. This new algorithm can also be used as an algorithm for the random generation of tableaux of a given shape with bounded entries. An appropriate deformation of this algorithm gives an algorithm for the random generation of plane partitions inside a given box
Orthogonal Decomposition and Packing of Complete Graphs

AbstractAn H-decomposition of a graph G is a partition of the edge-set of G into subsets, where each subset induces a copy of the graph H. A k-orthogonal H-decomposition of a graph G is a set of k H-decompositions of G, such that any two copies of H in distinct H-decompositions intersect in at most one edge. In case G=Kn and H=Kr, a k-orthogonal Kr-decomposition of Kn is called an (n, r, k) completely reducible super-simple design. We prove that for any two fixed integers r and k, there exists N=N(k, r) such that for every n>N, if Kn has a Kr-decomposition, then Kn also has an (n, r, k) completely-reducible super-simple design. If Kn does not have a Kr-decomposition, we show how to obtain a k-orthogonal optimalKr-packing of Kn. Complexity issues of k-orthogonal H-decompositions are also treated
Confirming Two Conjectures About the Integer Partitions

AbstractFor a given integer n, let Λn denote the set of all integer partitions λ1⩾λ2⩾…⩾λm>0 (m⩾1), of n. For the dominance order “⪯” on Λn, we show that if two partitions λ, μ are both chosen from Λn uniformly at random, and independent of each other, then Pr(λ⪯μ)→0 as n→∞. This statement answers affirmatively a question posed by Macdonald in 1979. The proof is based on the limit joint distribution of the largest parts counts found by Fristedt. A slight modification of the argument confirms a conjecture made by Wilf in 1982, namely that, for n even, the probability of a random partition being graphical is zero in the limit. The proof of the latter follows the footsteps of Erdo&#x030B;s and Richmond who saw that to confirm Wilf's conjecture it would be sufficient to show that the probability of the first k Erdo&#x030B;s–Gallai conditions of a partition being graphical approaches 0 as n, and then k approach infinity. The reason that the proofs of two seemingly unrelated conjectures turned out to be so close is that, as the E-R analysis revealed, the (joint) distribution of the largest part sizes in a partition λ and its dual λ′ coincides, in the limit, with the distribution of the largest part sizes for two independent partitions
On the Largest Caps Contained in the Klein Quadric of PG(5, q), q Odd

AbstractThis article studies the largest caps, of cardinality q3+q2+q+1, contained in the Klein quadric of PG(5, q), q odd. Presently, there are three examples of such caps known. They all are the intersection of the Klein quadric with a suitably chosen singular quadric with its vertex a line L and base a non-singular three-dimensional elliptic quadric. In this paper, we show that a (q3+q2+q+1)-cap contained in the Klein quadric of PG(5, q), q odd, q>3138, always is the intersection of the Klein quadric with another quadric, thus showing that such caps are a V43 variety of dimension 3 and of degree 4. This result also implies that a (q3+q2+q+1)-cap contained in the Klein quadric of PG(5, q), q odd, q>3138, defines a quadratic line complex of PG(3, q)
Applications of Minor-Summation Formula II. Pfaffians and Schur Polynomials

AbstractThe purpose of this paper is, to establish, by extensive use of the minor summation formula of pfaffians exploited in (Ishikawa, Okada, and Wakayama, J. Algebra183, 193–216) certain new generating functions involving Schur polynomials which have a product representation. This generating function gives an extension of the Littlewood formula. During the course of the proof we develop some techniques for computing sub-Pfaffians of a given skew-symmetric matrix. After the proof we present an open problem which generalizes our formula
Symplectic Analogs of the Distributive Lattices L(m, n)

AbstractWe introduce two families of symplectic analogs of the distributive lattices L(m, n). We give several combinatorial descriptions of these distributive lattices and use combinatorial methods to produce their rank generating functions. Using Proctor's sl(2, C) technique, we prove that these symplectic lattices are rank symmetric, rank unimodal, and strongly Sperner. This confirms a conjecture of Reiner and Stanton concerning one of these families of symplectic lattices. We describe how both families of symplectic lattices can be used to explicitly realize the fundamental representations of the symplectic Lie algebras
The Number of Ramified Coverings of the Sphere by the Double Torus, and a General Form for Higher Genera

AbstractAn explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to determine various linear recurrence equations for the numbers of these coverings with no ramification over infinity; one of these recurrence equations has previously been conjectured by Graber and Pandharipande. The general form of this series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two