Arnold Knopfmacher

On Carlitz Compositions

This paper deals with Carlitz compositions of natural numbers (adjacent parts have to be different). The following parameters are analysed: number of parts, number of equal adjacent parts in ordinary compositions, largest part, Carlitz compositions with zeros allowed (correcting an erroneous formula from Carlitz). It is also briefly demonstrated that so-called 1-compositions of a natural number can be treated in a similar style.

Graphs, partitions and Fibonacci numbers

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number >2^n^-^1+5 have diameter = ~. This is proved by using a natural correspondence between partitions of integers and star-like trees.

Gap-free compositions and gap-free samples of geometric random variables

We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise gap-free.

Combinatorics of geometrically distributed random variables: value and position of the r th left-to-right maximum

Abstract For words of length n , generated by independent geometric random variables, we consider the average value and the average position of the r th left-to-right maximum counted from the right , for fixed r and n →∞. This complements previous research (Discrete Math. 226 (2001) 255–267) where the analogous questions were considered for the r th left-to-right maximum counted from the left .

Counting polynomials with a given number of zeros in a finite field

Let denote a polynomial ring in an indeterminate X over a Unite field Exact formulae are derived for (i) the number of polynomials of degree n in with a specified number k of zeros in and (ii) the average number of zeros and corresponding variance for a polynomial of degree n in The main emphasis is on the case when multiplicity of zeros is counted.

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.

The number of distinct values in a geometrically distributed sample

For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (=letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word.

A SURVEY OF FACTORIZATION COUNTING FUNCTIONS

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distinct, or relatively prime, or all congruent to ±1 modulo 5. In this paper we review work on analogous problems concerning representations of n as a product of positive integers. We survey techniques for enumerating product representations both in the unrestricted case and in the case when the factors are required to be distinct, and both when the product representations are considered as ordered objects and when they are unordered. We offer some new identities and observations for these and related counting functions and derive some new recursive algorithms to generate lists of factorizations with restrictions of various types.

The exact length of the Euclidean algorithm in [ X ]

A study is made of the length L ( h, k ) of the Euclidean algorithm for determining the g.c.d. of two polynomials h , k in [ X ], a finite field. We obtain exact formulae for the number of pairs with a fixed length N which lie in a given range, as well as the average length and variance of the Euclidean algorithm for such pairs.

Combinatorics of geometrically distributed random variables: run statistics

For words of length n, generated by independent geometric random variables, we consider the mean and variance, and thereafter the distribution of the number of runs of equal letters in the words. In addition, we consider the mean length of a run as well as the length of the longest run over all words of length n.