Binomial posets, Möbius inversion, and permutation enumeration

Abstract

A unified method is presented for enumerating permutations of sets and multisets with various conditions on their descents, inversions, etc. We first prove several formal identities involving Möbius functions associated with binomial posets. We then show that for certain binomial posets these Möbius functions are related to problems in permutation enumeration. Thus, for instance, we can explain “why” the exponential generating function for alternating permutations has the simple form $\text{(}\text{1 + sin}\text{x)}\text{(}\text{cos}\text{x)}$. We can also clarify the reason for the ubiquitous appearance of e^x in connection with permutations of sets, and of ξ(s) in connection with permutations of multisets.