# Showing papers in "Journal of Combinatorial Theory, Series A in 1973"

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TL;DR: An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself and the structure and order of occurrence is universal for the class.

Abstract: An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself. Examples of four different types are studied in some detail; tables of numerical results are included. The limit sets are characterized by certain patterns; an algorithm for their generation is described and established. The structure and order of occurrence of these patterns is universal for the class.

518 citations

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TL;DR: A simple theorem on generating functions is proved which can be used to establish the asymptotic normality of an(k) as a function of k and local limit theorems are turned to in order to obtain asymPTotic formulas for an( k).

Abstract: Let a double sequence an(k) ⩾ 0 be given. We prove a simple theorem on generating functions which can be used to establish the asymptotic normality of an(k) as a function of k. Next we turn our attention to local limit theorems in order to obtain asymptotic formulas for an(k). Applications include constant coefficient recursions, Stirling numbers, and Eulerian numbers.

329 citations

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TL;DR: A drawing strategy is explained which applies to a wide class of combinatorial and positional games and when applied to n -dimensional Tic-Tac-Toe, it improves a result of Hales and Jewett.

Abstract: A drawing strategy is explained which applies to a wide class of combinatorial and positional games. In some settings the strategy is best possible. When applied to n -dimensional Tic-Tac-Toe, it improves a result of Hales and Jewett [5].

314 citations

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TL;DR: A construction is given for difference sets in certain non-cyclic groups with the parameters v, k, λ, n, which has minus one as a multiplier for every prime power q and every positive integer s.

Abstract: A construction is given for difference sets in certain non-cyclic groups with the parameters v = q s+1 {[ (q s+1 − 1) (q − 1) ] + 1} , k = q s (q s+1 − 1) (q − 1) , λ = q s (q s − 1) (q − 1) , n = q 2 s for every prime power q and every positive integer s . If q s is odd, the construction yields at least 1 2 (q s + 1) inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters ( v , k , λ , n ) = (4000, 775, 150, 625), which has minus one as a multiplier.

296 citations

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TL;DR: Knowing estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions.

Abstract: Known estimates of the maximal length of simple circuits in certain 3-connected planar graphs are surveyed and improved in several directions.

106 citations

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TL;DR: Questions of whether or not certain R are r -Ramsey where B is a Euclidean space and R is defined geometrically are investigated.

Abstract: The general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B . For a ϵ A denote by R ( a ) the set { b ϵ B | ( a , b ) ϵ R }. R is called r -Ramsey if for any r -part partition of B there is some a ϵ A with R ( a ) in one part. We investigate questions of whether or not certain R are r -Ramsey where B is a Euclidean space and R is defined geometrically.

104 citations

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TL;DR: The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.

Abstract: The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.

101 citations

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TL;DR: The object of this paper is to report that the number of codewords of weight 15 is zero, thus reducing the number-of-unknown weights to two.

Abstract: If a projective plane of order 10 exists, let A denote the (111, 56) binary error-correcting code generated by the rows of the incidence matrix. It is known that the weight distribution of this code is uniquely determined by the number of codewords of weights 12, 15 and 16. It is the object of this paper to report that the number of codewords of weight 15 is zero, thus reducing the number of unknown weights to two. Part of this calculation was carried out by computer.

80 citations

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TL;DR: A proof is given of the result that, if real n-dimensional Euclidean space Rn is covered by any n + 1 sets, then at least one of these sets is such that each distance d(0) is integers.

Abstract: A proof is given of the (known) result that, if real n-dimensional Euclidean space Rn is covered by any n + 1 sets, then at least one of these sets is such that each distance d(0

75 citations

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TL;DR: In this paper, it is shown that for the graph if λ = 1, then the graph exists for all υ such that ν ≢ 2 mod 3.

Abstract: Directed triple systems are an example of block designs on directed graphs. A block design on a directed graph can be defined as follows. Let G be a directed graph of k vertices which contain no loops. Let S be a set of υ elements. A collection of k-subsets of S with an assignment of the elements of each k-subset to the vertices of G is called a block design on G of order υ if the following is satisfied. Any ordered pair of elements of S is assigned λ times to an edge of G. For example, if S = {a, b, c, d, e} and and bae; cad; abc; dbe; acd; bce; adb; cde; aed; bec; is a collection of 3-subsets so written that in each subset the first element is assigned to the vertex 1, the second to 2, and the third to 3, then the collection is a block design on G with λ = 1. In this paper, it is shown that for the graph if λ = 1, then the graph exists for all υ such that ν ≢ 2 mod 3.

70 citations

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TL;DR: A method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementingSystem of m − 1 sequences.

Abstract: The following is proved (in a slightly more general setting): Let α1, …, αm be positive real, γ1, …, γm real, and suppose that the system [nαi + γi], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then α i α j is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all αi but one are integers; (iii) m ⩾ 5, two of the αi are integers, at least one of them prime; (iv) m ⩾ 5 and αn ⩽ 2n for n = 1, 2, …, m − 4. For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m − 1 sequences.

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TL;DR: The different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined and it is shown that one cannot always preassign the shape of a facet of a 4-polytope.

Abstract: The different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined. Using similar methods we show that one cannot always preassign the shape of a facet of a 4-polytope.

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TL;DR: A complete classification is given for neighborly 4-polytopes with 9 vertices and it is found that there are exactly 23 combinatorial types.

Abstract: A complete classification is given for neighborly 4-polytopes with 9 vertices. It is found that there are exactly 23 combinatorial types.

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TL;DR: It is given a counterexample to a conjecture of L. D. Baumert about zeros of copositive matrices and those matrices in E which are on extreme rays of the cone ofCopositiveMatrices which are positive semidefinite, respectively.

Abstract: Let E be the set of symmetric matrices in which every entry is 0 or ±1 and each diagonal entry is 1. We characterize those matrices in E which are, respectively, (a) copositive, (b) copositive-plus, (c) positive semidefinite. We characterize those copositive matrices in E which are on extreme rays of the cone of copositive matrices. We give a counterexample to a conjecture of L. D. Baumert about zeros of copositive matrices.

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TL;DR: For any prime p, the sequence of Catalan numbers a n = 1 n 2n−2 n−1 is divided by the an prime to p into blocks Bk(k > 0) of an divisible by p, whose lengths and positions are determined.

Abstract: For any prime p, the sequence of Catalan numbers a n = 1 n 2n−2 n−1 is divided by the an prime to p into blocks Bk(k > 0) of an divisible by p. The lengths and positions of the Bk are determined. Additional results are obtained on prime power divisibility of Catalan numbers.

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TL;DR: A new proof is given of the following theorem of Turyn: Let q = 2 n − 1 be a prime power ≡1 (mod 4); then there exists an Hadamard matrix of order 4 n that is of the Williamson type.

Abstract: In this paper a new proof is given of the following theorem of Turyn: Let q = 2 n − 1 be a prime power ≡1 (mod 4); then there exists an Hadamard matrix of order 4 n that is of the Williamson type.

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TL;DR: This paper determines the minimum number of quadruples such that each pair of elements of S is contained in at least one of them for all n divisible by 3.

Abstract: Let S be a finite set of order n. Let C(n, 4, 2) be the minimum number of quadruples such that each pair of elements of S is contained in at least one of them. In this paper C(n, 4, 2) is determined for all n divisible by 3. The case in which n is not divisible by 3 will be treated in a second paper.

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TL;DR: One consequence of this generating function is a formula which can be regarded as a q-analog of a well-known result arising in the representation theory of the symmetric group.

Abstract: The conjugate traceand traceof a planepartition aredefined,and thegenerating function for the number of plane partitions TT of n with

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Balliol College

^{1}TL;DR: There are at most three sets of extendable symmetric design parameters with any given value of λ, and the only twice-extendable asymmetric design is the 21-point projective plane.

Abstract: If a symmetric 2-design with parameters ( v , k , λ ) is extendable, then one of the following holds: v = 4 λ + 3, k = 2 λ + 1; or v = ( λ + 2)( λ 2 + 4 λ + 2), k = λ 2 + 3 λ + 1; or v = 111, k = 11, λ = 1; or v = 495, k = 39, λ = 3. In particular, there are at most three sets of extendable symmetric design parameters with any given value of λ. As a consequence, the only twice-extendable symmetric design is the 21-point projective plane.

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TL;DR: A common strengthening of the theorem that if A1, …, Al are subsets of a set S with n elements such that Ai ∩ Aj ≠ ∅ for all i, j then l ⩽ (k−1n−1) is proved.

Abstract: It was proved by Erdos, Ko, and Rado (Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 12 (1961) , 313–320.) that if A = {;A1,…, Al}; consists of k-subsets of a set with n > 2k elements such that Ai ∩ Aj ≠ ∅ for all i, j then l ⩽ (k−1n−1). Schonheim proved that if A1, …, Al are subsets of a set S with n elements such that Ai ∉ Aj, Ai ∩ Aj ≠ o and Ai ∪ Aj ≠ S for all i ≠ j then l ⩽ ( [ n 2 ] − 1 n − 1 ) . In this note we prove a common strengthening of these results.

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TL;DR: New results in the theory of non-serial dynamic programming are described and their computational relevance is pointed out.

Abstract: New results in the theory of non-serial dynamic programming are described in this paper. Their computational relevance is also pointed out.

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TL;DR: Let q be a number all whose prime factors divide integers of the form 2s − 1, s odd, if n = q + 2, the (3n) triples on n marks can be partitioned into q sets, each forming a Steiner triple system.

Abstract: Let q be a number all whose prime factors divide integers of the form 2s − 1, s odd. If n = q + 2, the (3n) triples on n marks can be partitioned into q sets, each forming a Steiner triple system.

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TL;DR: The problem is solved for |M| ⩽ 2, and some partial results are obtained in the general case.

Abstract: This paper deals with the following problem posed by Professor T. S. Motzkin: Suppose M is a given set of positive integers. How dense can a set S of positive integers be, if no two elements of S are allowed to differ by an element of M? The problem is solved for |M| ⩽ 2, and some partial results are obtained in the general case.

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TL;DR: A family T of k-subsets of an n-set such that no more than r have pairwise fewer than s elements in common is maximum (for sufficiently large n) only if T consists of all the k-sets containing at least one of r fixed disjoint s- Subsets.

Abstract: A family T of k-subsets of an n-set such that no more than r have pairwise fewer than s elements in common is maximum (for sufficiently large n) only if T consists of all the k-sets containing at least one of r fixed disjoint s-subsets.

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TL;DR: In case the structure is a generalized polygon, the irreducible representations of the algebra are computed and the theorem of Feit-Higman is deduced.

Abstract: We let correspond to any finite incidence structure S a certain semisimple algebra of endomorphisms of the vector space spanned by the flags of S. In case the structure is a generalized polygon we compute the irreducible representations of the algebra and deduce the theorem of Feit-Higman.

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TL;DR: Hall's theorem on distinct representatives is proved that there exists an infinite sequence on X with discrepancy at most 1 and this result is very close to the best possible.

Abstract: Let X be a finite or countable set and μ a measure on X with μ(X) = 1. Using Hall's theorem on distinct representatives we prove that there exists an infinite sequence on X with discrepancy at most 1. This result is very close to the best possible.

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TL;DR: The related combinatorial designs are the main concern of this paper; some results dealing with the existence and construction of modular Hadamard matrices will be included in a later paper.

Abstract: A square matrix with entries ± 1 is called a modular Hadamard matrix if the inner product of each two distinct row vectors is a multiple of some fixed (positive) integer. This paper initiates the study of modular Hadamard matrices and the combinatorial designs associated with them. The related combinatorial designs are the main concern of this paper; some results dealing with the existence and construction of modular Hadamard matrices will be included in a later paper.