Also see Corollary \[c:identity\] in Section \[s:ln\]. It is stated using a different notation involving lists.

3.  Among all [**cd**]{}-monomials of a given degree, which [**cd**]{}-monomial has the largest $\beta$ value?

    We treat this problem in Section \[s:maxima\]. Theorem \[t:maxima\] gives a simple and complete answer. Though the main idea of the proof is simple, we have to rely on two facts that are stated as exercises. This makes the proof a little unsatisfactory.

4.  What can be said about inequalities in general?

    A simple and striking inequality is provided by Lemma \[l:cc\_d\]. It says that replacing an occurrence of ${{\bf c}}^2$ by ${{\bf d}}$ in a ${{{\bf c}}{{\bf d}}}$-monomial increases its $\beta$ value. This is the first step for solving the question that was raised in item (3). Ehrenborg  has shown recently that this inequality holds for the ${{{\bf c}}{{\bf d}}}$-index of any polytope.

    There are two types of inequalities that we study in detail. The first type occur as portions of reverse unimodal sequences (Section \[s:unimodal\]) and the second type are the balance inequalities (Section \[s:balance\]). The motivation for the latter comes from a conjecture of Gessel about the [**ab**]{}-index of the Boolean lattice, which was proved in 