## Abstract

The notion of exponential Dowling structures is introduced, generalizing Stanley's original theory of exponential structures. Enumerative theory is developed to determine the Möbius 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 that the Möbius function is, up to a sign, the number of permutations in the symmetric group on r n + k elements having descent set {r, 2 r, ..., n r}. Using Wachs' original EL-labeling of the r-divisible partition lattice, the extended r-divisible partition lattice is shown to be EL-shellable.

Original language | English |
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Pages (from-to) | 311-326 |

Number of pages | 16 |

Journal | European Journal of Combinatorics |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2009 |

### Bibliographical note

Funding Information:The first author was partially supported by National Science Foundation grant 0200624. Both authors thank the Mittag-Leffler Institute where a portion of this research was completed during the Spring 2005 program in Algebraic Combinatorics. The authors also thank the referee for suggesting additional references.

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics