Among the unsolved problems in graphical enumeration listed in (4) is included the determination of the number of graphs and digraphs with a given partition. Parthasarathy (9) has developed a formulation for counting graphs with a given partition by making a suitable modification of the method given in (2) for the enumeration of graphs. We present here an analogous modification that leads to a formula for the number of digraphs with a given partition. Not surprisingly, the main combinatorial device for this purpose is provided by the classical theorem due to Pólya.

One of the many interesting conjectures proposed by S. M. Ulam in (5) can be stated as follows:

If G and H are two graphs with p points vi and ui respectively (p ⩾ 3) such that for all i, G — vi is isomorphic with H — ui then G and H are themselves isomorphic.

P. J. Kelly (3) has shown this to be true for trees. The conjecture is, of course, not true for p = 2, but Kelly has verified by exhaustion that it holds for all of the other graphs with at most six points. Harary and Palmer (2) found the same to be true of the seven-point graphs.

In (1) Harary reformulated the conjecture as a problem of reconstructing G from its subgraphs G — vi and derived several of the invariants of G from the collection G — vi.

A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property.

It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).

Suppose S = [a, b] is a number interval and F is a function from S X S to a normed algebraic ring N with multiplicative identity I. We consider the problem of finding, for appropriate conditions on F, a function M from S X S to N such that for all t and x,

where the integral is a Cauchy-left integral.

A graph is said to be d-polyhedral provided it is isomorphic with the graph formed by the vertices and edges of a d-dimensional bounded (convex) polyhedron (d-polyhedron). A k-tree is a connected acyclic graph in which each vertex is of valence ⩽k.

If is an n X n matrix, the permanent of A, Per A, is defined by

1

where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n!/nn. When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.

Block design games have been developed by Richardson (12) and by Hoffman and Richardson (8), who proved a number of theorems concerning such games by studying the number of elements in a blocking coalition. Hoffman and Richardson listed as unsolved (except for PG(2, 3)) the following problem: What is the minimum number of elements in a blocking coalition of a block design game?

This note considers blocking coalitions in those games that are dual to block designs having λ = 1 and r — k > 0. For such games certain blocking coalitions are shown to be related to sets of mutually disjoint blocks in the design to which the game is dual. In particular, for Steiner triple systems the largest odd-numbered set of mutually disjoint triples is shown to yield a minimum blocking coalition in the dual.

Let U be the closed disk in the plane, centred at the origin, and of unit radius. By a solid packing, or briefly a packing, C of U we shall understand a sequence ﹛Dn﹜, n = 1, 2, … , of open proper disjoint subdisks of U, such that the plane Lebesgue measures of U and of are the same. If rn is the radius of Dn and the complex number cn represents its centre, then the conditions for C to be a packing are

It was proved by Mergelyan (3) that for any packing the sum of the radii diverges:

1

Mergelyan's demonstration of (1) is somewhat involved and leans heavily on the machinery of functions of a complex variable. An elegant direct proof of (1) is given by Wesler (5), who uses the technique of projecting the boundaries of the disks of the packing on a diameter I of U.

Let

be a positive quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then f(x) assumes the value M for a finite number of integral vectors x = ±mk (k = 1 , … , s) called the minimal vectors.

Let f(z) be an entire function of the complex variable z = x + iy defined by the everywhere absolutely convergent Dirichlet series

1.1

If

then log m(x,f) is an increasing convex function of x (2), and

is called the Ritt order of f(z).

In (6) (see also 7), Sobolev introduced a class of function spaces Wm,p(Ω) (m a non-negative integer, 1 < p < ∞) defined on open subsets Ω of Euclidean space En, which have important applications in partial differential equations. They are defined as follows. For each n-tuple α = (α1, … αn) of non-negative integers let

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.

Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

In (3, §2), the writer and F. E. Browder stated briefly, without proof, some results concerning general distribution kernels. It is our aim here to prove and complete those results.

The terminology and notations are introduced in §1.

In §2 we define the notion of domain of dependence with respect to the kernel Kx,y (Definition 1) as well as the notion of smoothness of a distribution kernel at a point (Definition 2). Theorem 1 states that the set of points, where the distribution kernel is smooth, is open and the kernel is a smooth function in this set. Theorems 2 and 3 are the converse of Theorem 1.

Let A be a finite-dimensional (associative) algebra over an arbitrary field F. We shall say that a semi-group S is a translate of A if there exist an algebra B over F and an epimorphism ϕ: B → F such that A = 0→-1 and S = 1→-1. It is shown in (2) that any such semi-group S has a kernel (defined below) that is completely simple in the sense of Rees. Following Stefan Schwarz (4), we define the radical R(S) of S to be the union of all ideals I of S such that some power In of I lies in the kernel K of S. First we prove that the radical of a translate of A is a translate of the radical of A. It follows that A is nilpotent if and only if it has a translate S such that R (S) = S.

Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations

1.1

where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.

The following problem was raised by M. Brown. Let K be a finite simplicial complex, of dimension n, with αi(K) simplexes of dimension i. Which of the linear combinations have the property that they are unaltered by all stellar subdivisions of K? The most obvious invariant is the Euler characteristic; there are also some identities that hold for manifolds (2), so, if K is a manifold, they remain true on subdivision. We shall see that no other expressions are ever invariant, but if K resembles a manifold in codimensions ⩽2r (in a sense defined below) that r of the relations continue to hold.

If x- (i = 1, 2, … , 6) are homogeneous co-ordinates in [5] (the complex projective space of five dimensions), then the equation

1.1

represents the well-known (11) Perazzo cubic primalP43 of order 3 and dimension 4. With it is associated the Segre cubic threefoldS33 (12); specifically, S33 is the section of P43 by a tangent [4].

A net of degree k and order n is a set of n2 points and nk designated sets of points, called lines, such that

1. (1) The lines fall into k disjoint parallel classes, i.e. each line occurs in exactly one parallel class.

2. (2) Lines in the same parallel class have no points in common; lines in different parallel classes have exactly one point in common.

3. (3) Each point lies on exactly one line of each parallel class.

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebra A (i.e., one with no finite non-zero central projections), then the symmetries may be chosen from A. A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1 factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

Let ﹛Xj,j = 1, 2, 3, …﹜ be a sequence of independent, non-degenerate random variables and write

Under quite a diverse variety of conditions we may obtain

as n → ∞ for all x, — ∞ < x < ∞, and some real p ⩾ 0. For example, suppose the ﹛Xj﹜ happen to be distributed identically and belong to the domain of normal attraction of a symmetric stable law with characteristic exponent α, 0 < α ⩽ < 2, α ≠ 1.