c l}
\Psi(B_0) & = & {e}\\
\Psi(B_1) & = & 1 \\
\Psi(B_2) & = & {{\bf c}}\\
\Psi(B_3) & = & {{\bf c}}^2 + {{\bf d}}\\
\Psi(B_4) & = & {{\bf c}}^3 + 2 {{\bf c}}{{\bf d}}+ 2 {{\bf d}}{{\bf c}}\\
\Psi(B_5) & = & {{\bf c}}^4 + 3 ({{\bf c}}^2 {{\bf d}}+ {{\bf d}}{{\bf c}}^2) +
                5 {{\bf c}}{{\bf d}}{{\bf c}}+ 4 {{\bf d}}^2
     \end{array}$$

The Boolean lattice
-------------------

The Boolean lattice of rank $n+1$, which we denote by $B_{n+1}$, is the poset of all subsets of the set $[n+1]$ ordered by inclusion. It is same as the face lattice of the $n$-dimensional simplex. Hence it is an Eulerian poset and has a [**cd**]{}-index. Since the simplex is the simplest polytope, the Boolean lattice has a special role to play in the class of Eulerian posets. For example, among all Eulerian posets of rank $n+1$, the Boolean lattice $B_{n+1}$ has the smallest ${{{\bf c}}{{\bf d}}}$-index coefficient-wise. This was a conjecture of Stanley which was proved by Billera and Ehrenborg .

In this paper, we will be studying the Boolean lattice, not in relation to other posets, but rather as an object by itself. The flag $h$-vector (or 