as follows. Section \[sec:prelim\] contains some preliminaries. Our main results are Theorems \[theobnr\] and \[theobnr2\], whose particular case $r=1$ gives Stanley’s conjecture. The core of the proof is in Sections \[sec:stanbij\] and \[sec:inclexcl\], but it also relies on some combinatorial results on colored permutations which are in Sections \[sec:chro\], \[sec:genfunc\] and \[sec:bij\].

Acknowledgement {#acknowledgement .unnumbered}
===============

We thank Dominique Foata for his corrections and comments on this article.

Triangulations of the unit hypercube {#sec:prelim}
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This section contains nothing particularly new, but we introduce some notation and background (see [@beck; @stanley3]). Let $ \mathcal{X} \subset \mathbb{R}^n$ be a convex polytope with integral vertices. The [*Ehrhart polynomial*]{} $E(\mathcal{X},t)$ is defined as the unique polynomial in $t$ such that, for any integer $t>0$, $$\label{defehr}
  E( \mathcal{X} , t ) = \# \big(  t \mathcal{X} \cap \mathbb{Z}^n \big)$$ where $t\mathcal{X} := \{ tx \, : \, x\in\mathcal{X} \} $. The [*Ehrhart series*]{} of $\mathcal{X}$ is defined as $$\label{defehr2}
  E^* ( \mathcal{X}, z ) := (1-z)^{n+1} \sum_{t \geq 0} E( \mathcal{X}, t ) z^t.$$ From a general result of Stanley [@stanley3], the series $E^*(\mathcal{X},z)$ is in fact a polynomial with positive integral coefficients. As it often happens, it is an interesting problem to find 