Research in Enumerative Combinatorics

Brandeis University, Waltham MA

Investigators

Abstract

The investigator studies problems in enumerative combinatorics related to permutations, symmetric functions, and lattice paths. One of the investigator's approaches is to apply Richard Stanley's theory of P-partitions and its generalizations to the study of descent algebras of symmetric and related groups. Several other problems related to the enumeration of permutations by descents are also studied. The investigator also studies a generalization of two-stack-sortable permutations leading to a symmetric-function refinement of the formula of West (proved by Zeilberger) for counting these permutations, and several problems involving the enumeration of lattice paths. Enumerative combinatorics involves counting mathematical objects with given properties. It has applications to computer science, physics, and chemistry, as well as to other areas of mathematics, such as algebra, probability and statistics, topology, analysis, and number theory. The investigator's long-range goal is to develop, extend, and systematize the methods of enumerative combinatorics, and the problems studied will contribute towards this goal.