Combinatorics

Dr. John Konvalina's recent research results were the subject of a combinatorics seminar held at Cal Tech (California Institute of Technology) in April 2000 and presented by Professor Bruce Rothschild of UCLA under the title Konvalina's generalization of the binomial coefficient and the subset-subset analogy. These results will appear in the December 2000 issue of the American Mathematical Monthly under the title A Unified Interpretation of the Binomial Coefficients, the Stirling Numbers, and the Gaussian Coefficients.

The binomial coefficients can be used to count the number of ways to choose objects from a set where the order of the objects does not matter. For example, the number of ways to form a committee of 5 people chosen from a department with 20 faculty members can be expressed as a binomial coefficient. The Stirling numbers count set partitions. For example, the number of ways to divide a classroom of 30 students into 5 nonempty groups can be expressed as a Stirling number. The Gaussian coefficients arise when counting certain matrices or two-dimensional arrays (technically called row-reduced echelon form matrices). These three classes of numbers have other interpretations and arise in many branches of mathematics, including discrete mathematics, number theory, linear algebra, and geometry. Despite the different interpretations, these numbers have strikingly similar properties. Dr. Konvalina found a simple unifying interpretation involving choosing balls with or without repetition from weighted boxes. In this model the similarities of these numbers can be derived in a single stroke.

Dr. John Konvalina

(Editor's note: Professor Konvalina came to UNO 1975.)