Abstract
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.
Highlights
The set partitions of [n] = {1, 2, · · ·, n} (denoted (n)) are a classical object of combinatorics
In studying the character theory of upper triangular matrices we were led to some unusual statistics on set partitions
How does d(λ) vary with λ? As shown below, its mean and second moment are determined in terms of the Bell numbers Bn d(λ) = − 2Bn+2 + (n + 4)Bn+1 λ∈ (n) d2(λ) =4Bn+4 − (4n + 15)Bn+3 + (n2 + 8n + 9)Bn+2 − (4n + 3)Bn+1 + nBn
Summary
This paper gives a large family of statistics that admit similar formulae for all moments These include classical statistics such as the number of blocks and number of blocks of size i. Section ‘Computational results’ gives computational results; determining the coefficients in shifted Bell expressions involves summing over all set partitions for small n. Statement of the main results Let (n) be the set partitions of [ n] = {1, 2, · · · , n} (so | (n)| = Bn, the nth Bell number). A statistic on λ is defined by counting the number of occurrences of patterns. (iii) A simple statistic is defined by a pattern P of length k and Q ∈ Z[ y1, · · · , yk, m]. F(P) is the set of firsts elements, L(P) is the set of lasts, A is the arc set of the pattern, and C(P) is the set of consecutive elements
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