(-1)^{\rho(x,y)}$, where $\mu$ denotes the Mobius function of the interval $(x,y)$ of $P$ and where $\rho(x,y) = \rho(y) - \rho(x)$. Here $\rho$ is the rank function of $P$. An important example of an Eulerian poset is the face lattice of a convex polytope.

If $P$ is Eulerian then $\Psi_P({{\bf a}},{{\bf b}})$ can be written uniquely as a polynomial $\Psi_P({{\bf c}},{{\bf d}})$ in the non-commuting variables ${{\bf c}}= {{\bf a}}+{{\bf b}}$ and ${{\bf d}}= {{\bf a}}{{\bf b}}+{{\bf b}}{{\bf a}}$.

This fact was noticed by Fine and proved by Bayer and Klapper, see  or . It is equivalent to the statement that the linear relations satisfied by the flag $h$-vector of an Eulerian poset are precisely the generalised Dehn-Sommerville equations, also known as the Bayer-Billera relations .

The polynomial $\Psi_P({{\bf c}},{{\bf d}})$ is called the [**cd**]{}-index of $P$. Henceforth we will just refer to it as $\Psi(P)$. Note that the degree of $\Psi(P)$ is one lower than the rank of the poset $P$. (The variables ${{\bf c}}$ and ${{\bf d}}$ are assigned degrees $1$ and $2$ respectively.) It also follows from the definition that for a poset $P$ of rank $n+1$, the coefficient of ${{\bf c}}^n$ in $\Psi(P)$ is always $1$.

$$\begin{array}{r 