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Showing papers in "Journal of Combinatorial Theory, Series A in 1972"


Journal ArticleDOI
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.

1,029 citations


Journal ArticleDOI
Joseph B. Kruskal1
TL;DR: The purpose of this paper is to describe the intriguing subject of well-quasi-ordering, and to illustrate the theory, here are two definitions and an elementary result.

367 citations


Journal ArticleDOI
TL;DR: The general theorems presented here will be illustrated by examples and will be applied in the second part of this article, “An Existence Theory for Pairwise Balanced Designs, II.”

218 citations


Journal ArticleDOI
TL;DR: A new upper bound is obtained for φ(2, k) for all k ⩾ 3 and a new lower bound forπ(n, 3) is obtained, both for positive integers n and k.

217 citations



Journal ArticleDOI
TL;DR: Using some recent results involving Young tableaux and matrices of non-negative integers, it is possible to enumerate various classes of plane partitions by actual construction by simple proof of MacMahon's generating function for plane partitions.

200 citations


Journal ArticleDOI
TL;DR: It is shown that the conditions (1) and (2) are both necessary and sufficient for the existence of a (v, k, A)-BIBD when k = 3,4 and for all A.

183 citations


Journal ArticleDOI
TL;DR: A short proof of the lattice property of (Tn, →), the system of binary bracketings ordered by a semi-associative law, is presented.

137 citations


Journal ArticleDOI
TL;DR: Doubly regular tournaments and homogeneous tournaments are defined and shown to be equivalent and existence of such a tournament is equivalent to the existence of a skew Hadamard matrix of order n + 1.

106 citations


Journal ArticleDOI
TL;DR: It is shown that P leaves the (2q + 2, q + 1) code in the family invariant, and that P{I, −I} is isomorphic to PGL2(q), and that a Hadamard matrix is left invariant by the group P described above.

102 citations


Journal ArticleDOI
Richard J Turyn1
TL;DR: An infinite family is constructed here, and as a corollary it is shown that an Hadamard matrix of order 6( q + 1) exists if q is a prime power ≡ 1 (mod 4).

Journal ArticleDOI
P Mani1
TL;DR: All n -dimensional spheres are realizable as boundary complexes of polytopes as well as simplicial n -sphere with few vertices if their number does not exceed n + 3.

Journal ArticleDOI
TL;DR: The central result of this paper is a generalization of the theorem that, for n ≥ 5, every even permutation defined on n symbols is a commutator a b a−1 b−1 of even permutations a and b.

Journal ArticleDOI
TL;DR: In both cases the cycle structures induced on cyclic permutations by disjoint transpositions or the decomposition of a statecycle by interchanging the successors of disJoint state pairs are investigated.

Journal ArticleDOI
TL;DR: Necessary existence conditions for the existence of a balanced array ( B -array) (“partially balanced” array) of 2 symbols, strength t , and m rows (with m ⩽ t + 2).

Journal ArticleDOI
David C Kurtz1
TL;DR: In this note sufficient conditions are given to guarantee that an array will have the SLC property, making use only of the recurrence (1).

Journal ArticleDOI
TL;DR: Here the authors construct a 5-(2n + 2, 2n−1 + 1, λ) design for every n ⩾ 4, with λ = (2 n−1 − 3)(2n−2 − 1).

Journal ArticleDOI
TL;DR: It is proved that for balanced incomplete block designs with blocks having five elements each the known necessary existence condition is also sufficient, with the exception of the non-existing design λ = 2, v = 15.

Journal ArticleDOI
TL;DR: An algorithm is presented for the generation of the Ford sequence of length 2n which does not require storage of the sequence and it is shown that the sequence does not need to be stored.

Journal ArticleDOI
TL;DR: The question whether there exists a first row in such a way that half of all signs of the triangle is positive, is answered in the affirmative.

Journal ArticleDOI
TL;DR: Abstract Soient k un entier positif, X un ensemble et ϕ un homomorphisme du semigroupe libre engendre par X dans le produit direct du groupe cyclique infini par un semi-groupe fini.

Journal ArticleDOI
TL;DR: In this paper, the authors studied multiplicative designs that are characterized by a (0, 1)-matrix A of order n ⩾ 3 that satisfies the matrix equation.



Journal ArticleDOI
TL;DR: This paper proves the following result and transfinite extensions of it: If the cardinalities |I| = f and |S| = n are finite and f > n(r − 1), then one can find r disjoint subsets Iυ(υ = 1,…,r) of I for which ⋃ i∈I 1 M i = … = ⋁ i∈�I r M i

Journal ArticleDOI
TL;DR: The present paper is of a combinatorial nature and does not use the Perron-Frobenius theorem, but it is well known that a matrix A has total support if and only if there exists a doubly stochastic matrix of the same size as A having the same pattern as A.

Journal ArticleDOI
TL;DR: It is shown that if n ≡ 0 mod 4, then the existence of such a rotation on n partnerships implies theexistence of an n × n Hadamard matrix, and the method succeeds in designing complete, balanced Howell rotations whenever n − 1 is a prime-power.

Journal ArticleDOI
TL;DR: A proof is given for the novel identity q (a−k)(b−k) due to Richard Stanley, where m = min(a, b), and brackets denote q-binomial coefficients defined by x n = Φ j=1 n q x−j+1 −1 q j −1, [ x 0 ] = 1.

Journal ArticleDOI
TL;DR: This paper studies the problem when G is a p-group and an explicit generating function for the number of transformation groups in the case when G has prime order.

Journal ArticleDOI
TL;DR: It is conjectured that every Hadamard matrix of order mn is decomposable into m components for m = 4 or 8 and the decomposition is defined.