This award supports the research in combinatorics of Professor Ira Gessel of Brandeis University. Dr. Gessel plans to study the applications of symmetric functions and orthogonal polynomials to problems in enumerative combinatorics. Among the topics involved are coefficient extraction for symmetric functions, permutation and sequence enumeration, Lagrange inversion, Polya frequency sequences, rook polynomials, combinatorial interpretations of integrals involving orthogonal polynomials, and orthogonal polynomials associated with Genocchi numbers. This research falls in the broad category of combinatorics, which is one of the most active fields in today's mathematics. Fundamentally, combinatorics represents a systematization of the very first of all mathematical activities, counting. In its modern development, however, combinatorics has gone beyond just counting to make use of a wide variety of advanced mathematical techniques, and although its roots go back several centuries, the field has had an explosive development in the past few decades because of its importance in communications and information technology.
