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Closed expressions for averages of set partition statistics. (English) Zbl 1339.15019
Summary: 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 \([n]=\{1,2,\dots,n\}\) 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 \(n\).

MSC:
15B33 Matrices over special rings (quaternions, finite fields, etc.)
05A15 Exact enumeration problems, generating functions
11B73 Bell and Stirling numbers
15A30 Algebraic systems of matrices
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