We consider the enumeration of partitions of a finite set according to the number of consecutive elements inside a block under the assumption that the elements are arranged around a circle. This statistic, commonly known as circular succession, continues to play a significant role in many combinatorial problems involving combinations of a set following its first appearance in a paper of Irving Kaplansky in the 1940s. In this paper we obtain interesting formulas for the number of partitions avoiding a circular succession and the number of partitions containing a specified number of circular successions. Our methods include both elementary combinatorial reasoning and the application of ordinary and exponential power series generating functions. Several new combinatorial identities are also stated.
Bibliographical noteFunding Information:
The authors are grateful to the anonymous referees for constructive comments which led to improvements in exposition. The second author was partially supported by the National Research Foundation of South Africa under grant number 80860 .
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics