Balls in Boxes: Variations on a Theme of Warren Ewens and Herbert Wilf

Computing the distribution of the maximum in balls-and-boxes problems with application to clusters of disease cases.

We present a rapid method for the exact calculation of the cumulative distribution function of the maximum of multinomially distributed random variables. The method runs in time O(mn), where m is the desired maximum and n is the number of variables. We apply the method to the analysis of two situations in which an apparent clustering of cases of a disease in some locality has raised epidemiolog...

There's plenty of time for evolution.

Objections to Darwinian evolution are often based on the time required to carry out the necessary mutations. Seemingly, exponential numbers of mutations are needed. We show that such estimates ignore the effects of natural selection, and that the numbers of necessary mutations are thereby reduced to about K log L, rather than K(L), where L is the length of the genomic "word," and K is the numbe...

For Kathy and Elisabetta

A Survey of Combinatorial Gray Codes

The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some pre-specified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960’s and 70’s on min...

Regenerative Partition Structures

A partition structure is a sequence of probability distributions for πn, a random partition of n, such that if πn is regarded as a random allocation of n unlabeled balls into some random number of unlabeled boxes, and given πn some x of the n balls are removed by uniform random deletion without replacement, the remaining random partition of n − x is distributed like πn−x, for all 1 ≤ x ≤ n. We ...