Amram Meir

On packing of squares and cubes

Abstract The main result of the paper is the following: Suppose x 1 ≥ x 2 ≥… are the sides of cubes in the k -dimensional space and a 1 , a 2 ,…, a k are the edges of a rectangular parallelepiped. It is possible to pack the cubes into the parallelepiped if a j ≥ x 1 , j =1,2,…, k and x 1 k + ∏ j = 1 k ( a j − x 1 ) ⩾ V where V denotes the volume of the cubes.

On an asymptotic method in enumeration

Abstract A number of enumeration problems involving tree-like structures lead to a generating function w = y(z) that satisfies a functional relation w = F(z, w). The value of r, the radius of convergence of y(z), provides information about the behaviour of the coefficients of y(z). Sufficient conditions are given here for determining r and also for ensuring that z = r is the only singularity on the disk |z|⩽r.

The distance between points in random trees

Abstract Let γ denote the number of points in the path joining two arbitrary points in a random tree T n with n labeled points. It is shown, among other things, that E(γ)∼(1/2πn) 1/2 .

On maximal independent sets of nodes in trees

A subset / of the nodes of a graph is a maximal independent set if no two nodes of / are joined to each other and every node not in / is joined to at least one node in /. We investigate the behavior of the average number e(n) and the average size μ(n) of maximal independent sets in trees Tn where the averages are over all trees Tn belonging to certain families of rooted trees. We find, under certain conditions, that e(n) ∼ q · En and μ(n) ∼ Sn as n ∞, where q, E, and S are constants that depend on the family being considered.

The asymptotic behaviour of coefficients of powers of certain generating functions

We investigate the asymptotic behaviour of the coefficients in powers of generating functions y (x) that satisfy a relation of the form y = x φ (y).

Hereditarily finite sets and identity trees

Abstract Some asymptotic results about the sizes of certain sets of hereditarily finite sets, identity trees, and finite games are proven.

Packing and covering constants for certain families of trees. I

If ℱ denotes a family of rooted trees, let pk(n) and ck(n) denote the average value of the k-packing and k-covering numbers of trees in ℱ that have n nodes. We assume, among other things, that the generating function y of trees in ℱ satisfies a relation of the type y = xϕ(y) for some power series ϕ. We show that the limits of pk(n)/n and ck(n)/n as n ∞ exist and we describe how to evaluate these limits.

On random mapping patterns

Random mapping patterns may be represented by unlabelled directed graphs in which each point has out-degree one. We determine the asymptotic behaviour of various parameters associated with such graphs, such as the expected number of points belonging to cycles and the expected number of components.

On total matching numbers and total covering numbers of complementary graphs

Abstract Best upper and lower bounds, as functions of n, are obtained for the quantities β 2 (G)+β 2 ( G ) and α 2 (G)+α 2 ( G ) , where β2(G) denotes the total matching number and α2(G) the total covering number of any graph G with n vertices and with complementry graph Ḡ. The best upper bound is obtained also for α2(G)+β2(G), when G is a connected graph.

A Tauberian theorem concerning weighted means and power series

Suppose throughout that { p n } is a sequence of non-negative numbers with p 0 > 0, that and that Let { s n } be a sequence of real numbers.