Exponential Dowling structures

Details Exponential Dowling structures Exponential Dowling structures

2010-09-21

arxiv.org/abs/1009.4202v1

European J Combinatorics, 30 (2009), 311--326

Mathematics

Combinatorics

17 pages

Scientific paper

The notion of exponential Dowling structures is introduced, generalizing Stanley's original theory of exponential structures. Enumerative theory is developed to determine the M\"obius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley's study of permutations associated with exponential structures leads to a similar vein of study for exponential Dowling structures. In particular, for the extended r-divisible partition lattice we show the M\"obius function is, up to a sign, the number of permutations in the symmetric group on rn+k elements having descent set {r, 2r, ..., nr}. Using Wachs' original EL-labeling of the r-divisible partition lattice, the extended r-divisible partition lattice is shown to be EL-shellable.

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