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Open Access Research

Closed expressions for averages of set partition statistics

Bobbie Chern1, Persi Diaconis2, Daniel M Kane3 and Robert C Rhoades4*

Author Affiliations

1 Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA

2 Department of Mathematics and Statistics, Sequoia Hall, 390 Serra Mall, Stanford University, Stanford, CA 94305, USA

3 Department of Mathematics, Bldg 380, Stanford University, Stanford, CA 94305, USA

4 Center for Communications Research, 805 Bunn Dr., Princeton, NJ 08540, USA

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Research in the Mathematical Sciences 2014, 1:2  doi:10.1186/2197-9847-1-2

Published: 17 June 2014

Abstract

In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field, we found that the moments (mean, variance, and higher moments) of novel statistics on set partitions of [n]={1,2,⋯,n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulae for all n.