Exponential Dowling structures

TL;DR

This paper introduces exponential Dowling structures, extending Stanley's exponential structures, and develops their enumerative properties, including M"obius functions and shellability, with applications to permutation enumeration.

Contribution

It generalizes Stanley's exponential structures to exponential Dowling structures and explores their combinatorial and topological properties, including M"obius functions and EL-shellability.

Findings

M"obius function related to permutation counts

Extended r-divisible partition lattice is EL-shellable

Enumeration formulas for permutations with specific descent sets

Abstract

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.