Robert E. Greenwood

Combinatorial relations and chromatic graphs

1. Introduction. The following elementary logical problem was a question in the William Lowell Putnam Mathematical Competition held in March 1953 (1): Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn, and then painted, some segments red, some blue. Prove that some triangle has all its sides the same color.

The Number of Cycles Associated with the Elements of a Permutation Group

Gontcharoff [1] has given several moment generating functions for the number of cycles associated with the elements of the permutation group on N symbols. Recently the author rediscovered the factorial moment generating function for the above problem previously given by Gontcharoff. Since the method of derivation is quite different from that given by Gontcharoff, it may be worth reproducing here. For each positive integer N, we let GN denote the group of N! permutations on N symbols (which we may refer to as being cards numbered from 1 to N for convenience). As is well known, each of these permutations may be written as a product of disjoint cycles of lengths ti, t2, * , t. say, and these lengths will satisfy the relation Eti =N. We call the permutation (1, 2, * , N) the standard permutation. The factorial moment generating function found by the author is based on the form

A birthday holiday problem

A company hires employees without regard to their birth dates, and then gives all employees a holiday on the birthday anniversary of each employee. What is the number of employees that shoul be hired to obtain the maximum number of man-days during the year? It is shown that, for a year with g days, g >1, the number of employees to be hired should be either g −1 or g .

An exact probability distribution for a card matching problem

Abstract Dedicatory Statement . Professor H. S. Vandiver on several occasions has told the author of G. H. Hardys acknowledgment of indebtedness to the numerical calculations made by Major P. A. MacMahon and of other significant tabular studies used in testing and formulating a theory. MacMahon computed the exact values of the partition numbers p ( n ) for n = 1, 2, ···, 200 at the request of Hardy, and these values were used by Hardy and Ramanujan to test their first approximating formula for p ( n ) and later to devise other and better approximating formulas for p ( n ). Some indications of the indebtedness of Hardy and Ramanujan to this numerical work of Major MacMahon can be found in references [1], [2], and [3]. Professor Vandiver himself has expressed his own ideas of the importance of numerical tables in the foreword of “Tables of All Primitive Roots of Odd Primes Less Than 1000” by Roger C. Osborn [4]. This short table of exact numerical probabilities is therefore respectfully dedicated to Professor Harry S. Vandiver.