Joel Stein

In 1964 Gian-Carlo Rota initiated a new school of algebraic combinatorics with the publication of the first installment of his Foundations of Combinatorial Theory series of papers. Rota and his school of young disciples, including Crapo, Stanley, Doubilet and Goldman began the serious work of reformulating much of traditional combinatorial theory from a modern point of view. One classic triumph of this work, developed by Gian-Carlo Rota and Peter Doubilet, was an elegant new approach to the theory of symmetric functions. Using a new set of tools recently introduced into algebraic combinatorics by Doubilet, Rota and Stanley, they succeeded in casting the classic theory of symmetric functions in a new combinatorial light. One of the consequences of this approach was Peter Doubilet's essentially new theory of symmetric functions indexed by the partitions of a set. It was published by him in "On the foundations of combinatorial theory. VII. Symmetric functions through the theory of distribution and occupancy", Studies in Appl. Math. 51 (1972), 377-396. This paper stands today as one of the truly perfect papers in modern algebraic combinatorics. The full force of Rota's combinatorial philosophy of generating functions and Mobius functions is brought to bear in this remarkable paper containing many new and original ideas.

In this talk, we return to Doubilet's paper for a fresh look at some of the constructions. Rather than using Doubilet's approach with generating functions for sets of mappings, we will develop the main ideas using some simple constructions from the theory of vector symmetric functions. Doubilet's theory of symmetric functions indexed by the partitions of a set is cast in a new light. We reveal that it is just the theory of vector symmetric functions of degree at most one in each of the vector coordinates! By imitating one of Doubilet's constructions, we will introduce some new classes of symmetric functions indexed by segments in the lattice of partitions of a set. They share a multiplicative property pointed out and investigated by Doubilet in Foundations VII. Doubilet's treatment of the Kronecker Inner Product for his new symmetric functions is also cast in a new light, it is just the Mobius Algebra for the lattice of partitions of a set! We will also introduce a vector analogue of the Frobenious map from permutations to symmetric functions, adapted to the Doubilet theory of symmetric functions indexed by the partitions of a set. Along the way, some key combinatorial lemmas about Mobius functions will be reinterpreted with surprising consequences.