3:45PM: A filtered algebra for set partitions
Persi DiaconisThis talk is about some polynomial algebras that arise in enumerative combinatorics. A set partition is a partition of an 1,2,...,n into blocks (thus there are 5 set partitions of 3: 1/2/3 12/3 13/2 123) the number of set partitions of n is the Bell number B(n) (So B(3) = 5). In studying the character theory of the uni upper triangular group with elements in a finite field, a variety of natural statistics arise. Things like the number of blocks, the number of crossings and the sum of the largest - smallest numbers in each block. All of these had averages which have a simple form, sums of a few terms P(k,n) B(n+k) where P(k, ) is a polynomial in n. If this is known apriori, then one can compute the exact result for small n, solve for the coefficients of P(k, ) and know the formula for all n. In joint work with Bobbie Chern,Daniel Kane and Rob Rhodes we introduce an algebras of 'statistics' and show that the averages (indeed all the moments) are shifted Bell polynomials. The Algebra is filtered and I hope that this audience will help me understand it better. I promise to keep the exposition 'friendly' No specialist combinatorial knowledge required (If you want to look, the paper(s) are on the Arxiv and on my home page (type persi into google).