number of excedences. Nan Li also extended the result to colored permutations by considering the hypercube $[0,r]^n$ for some integer $r>0$, and the polytopes $\{ v\in [0,r]^n \,:\, k\leq \sum v_i \leq k+1 \}$ are the multi-hypersimplices referred to in the title of this article. We would like to mention that besides Stanley’s conjecture, some recent works deals with the geometry and combinatorics of hypersimplices, see [@hibi; @ohsugi].

The goal of this article is to give another generalization of Stanley’s conjecture to colored permutations. Our result is stated in terms of the flag descents and flag excedences in colored permutations, and relies on some related work by Foata and Han [@foatahan]. Our method gives in particular a new proof of Stanley’s conjecture in the uncolored case. Our method can roughly be described as follows. We first consider the case of the half-open hypercube $[0,r)^n$, where an analog of Stanley’s conjecture in terms of descents and inverse descents can be proved in a rather elementary way. We can relate the half-open hypercube $[0,r)^n$ with the usual hypercube $[0,r]^n$ via an inclusion-exclusion argument. Then, it remains only to prove an identity relating two generating functions for colored permutations.

This article is organized 