EAPSI: A Unified Approach to the Euler-Mahonian Identity

Olsen Mccabe J, Lexington KY

Investigators

Abstract

A permutation of a finite set is a reordering of the elements of the set. Permutation statistics, such as descent number and major index, describe the behavior of a particular permutation. The Euler-Mahonian identity is a mathematical identity related to these permutation statistics which arises in many distinct areas of mathematics. This project will investigate the underlying connections between the different mathematical areas by identifying connections between the various proofs of the identity. The identification of such connections will allow for a more standard approach when studying other, often more difficult, mathematical identities of this type. Research will be conducted with Professor Takayuki Hibi, an expert in combinatorial mathematics, at Osaka University. More explicitly, the Euler-Mahonian identity is a multivariate generating function identity involving the joint distribution of the descent number and major index over symmetric group Sn. The researcher plans to investigate connects between three proofs of the identity, namely (1) a method involving counting integer points in convex/discrete geometry, (2) a method involving commutative algebraic and representation theoretic techniques in the context of the coinvariant algebra for the symmetric group Sn, and (3) a method involving finding ordinary generating functions arising from lecture hall partitions. The identification of such connections will allow for extending to similar, but more complicated combinatorial identities of this type, such as those involving statistics of the hyperoctohedral group or colored permutation groups. Professor Hibi, along with his research group at Osaka University, are leaders in algebraic combinatorics and combinatorial commutative algebra. Subsequently, Professor Hibi?s insight and expertise will be an invaluable aid in developing a unified approach to generating functions of this type. This award under the East Asia and Pacific Summer Institutes program supports summer research by a U.S. graduate student and is jointly funded by NSF and the Japan Society for the Promotion of Science.