Theresa P. Vaughan

Enumeration of pairs of sequences by rises, falls and levels

The paper is concerned with the enumeration of pairs of sequences with given specification according to rises, falls and levels. Thus there are nine possibilities RR, ..., LL. Generating functions in the general case are very complicated. However in a number of special cases simple explicit results are obtained.

Whitney numbers of the second kind for the star poset

The integers W0, ..., Wt are called Whitney numbers of the second kind for a ranked poset if Wk is the number of elements of rank k. The set of transpositions T = {(1, n), (2, n), ..., (n - 1, n)} generates Sn, the symmetric group. We define the star poset, a ranked poset the elements of which are those of Sn and the partial order of which is obtained from the Cayley graph using T. We characterize minimal factorizations of elements of Sn as products of generators in T and provide recurrences, generating functions and explicit formulae for the Whitney numbers of the second kind for the star poset.

Families Implying the Frankl Conjecture

A union closed (UC) family A is a finite family of sets such that the union of any two sets in A is also in A. Peter Frankl conjectured that for every union closed family A, there exists some x contained in at least half the members of A. An FC-family is a UC family B such that for every UC family A, if B?A, then A satisfies the Frankl conjecture. We construct several such families, involving five, six, and seven elements.

On Union-Closed Families, I

A union closed familyAis a finite family of sets such that the union of any two sets inAis also inA. The conjecture under consideration is Conjecture 1: For every union closed familyA, there exists somexcontained in at least half the members ofA. We study the structure of such families (as partially ordered sets), and verify the conjecture for a large number of cases.

Cycles of directed graphs defined by matrix multiplication (mod n )

Abstract Let A be a k × k matrix over a ring R; let GM( A , R ) be the digraph with vertex set R k , and an arc from v to w if and only if w = Av . In this paper, we determine the numbers and lengths of the cycles of GM( A , R ) for k =2 in the following two cases. (a) R= F q , the q-element finite field, and (b) R= Z /n Z and GCD( n ,det( A ))=1. This extends previous results for k =1 and R= Z /n Z . We make considerable use of the Smith form of a matrix; other than that, the most powerful tool we use is the Chinese Remainder Theorem.

Counting and constructing orthogonal circulants

Abstract If F is an arbitrary finite field and T is an n × n orthogonal matrix with entries in F then one may ask how to find all the orthogonal matrices belonging to the algebra F[T] and one may want to know the cardinality of this group. We present here a means of constructing this group of orthogonal matrices given the complete factorization of the minimal polynomial of T over F. As a corollary of this construction scheme we give an explicit formula for the number of n × n orthogonal circulant matrices over GF(pl) and a similar formula for symmetric circulants. These generalize results of MacWilliams, J. Combinatorial Theory10 (1971), 1–17.

Cycles of linear permutations over a finite field

Abstract We study the cycle structure of those permutations of the finite field Fqn of the form L(x) = ∑ i=0 n−1 a i x q i where each ai ϵ Fq. For such a permutation, the problem of finding its cycle decomposition of Fqn can be reduced to finding its cycle decomposition on certain T-invariant subspaces of Fqn, where T is the operator defined by T : x → xq. If L1(x) and L2(x)M are in the above form, we say that L1(x) and L2(x) are equivalent if L1(x) and L2(x) induce the same cycle decomposition of Fqn, and we say they are strongly equivalent if they induce the same cycle decomposition in every T-invariant subspace of Fqn. We show that these notions are not the same, and we give characterizing theorems for each.

The discriminant of a quadratic extension of an algebraic field

Let F be an algebraic field, and K an extension of F of degree 2. We describe a method for computing the relative discriminant D for K over F. We work out the details for the case when F is quadratic and give tables which yield D very easily. We also apply the method to one type of cubic field F, and give tables for it.

On computing the discriminant of an algebraic number field

Letf(x) be a monic irreducible polynomial in Z[x], and r a root of f(x) in C. Let K be the field Q(r) and X the ring of integers in K. Then for some k E Z, disc r = k2 disc M. In this paper we give constructive methods for (a) deciding if a prime p divides k, and (b) if p I k, finding a polynomial g(x) E Z[x] so that g(x) i 0 (mod p) but g(r)/p E M.

A group of integral points in a matrix parallelepiped

Abstract Let A be an n × n integral matrix with determinant D >0, and let P ( A ) be the n -parallelepiped determined by the columns { A i } n i =1 of A , P(A)= ∑ i=1 n x i A i 0 i Let L be the set of integral vectors in P ( A ), and let G ( A ) be the subset of L consisting of vectors whose coefficients x i satisfy 0⩽ x i G ( A ), equipped with addition modulo 1 on the coefficients x i , is an Abelian group of order D , whose invariant factors are the invariant factors of the integral matrix A . We give a formula for | L |, and show that | L | is not a similarity invariant.