is a polynomial with positive integer coefficients. We refer the reader to the basic paper of Stanley . For more recent references, see . In this section, we first review the basic definitions and then motivate the problem that we study.

Background
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Let $P$ be a graded partially ordered set (poset) of rank $n+1$. For $S$, a subset of $[n]$, let $f_{S}$ be the number of chains (flags) in $P$ that have elements on the ranks in $S$. These $2^n$ numbers constitute the flag $f$-vector of the poset $P$. The flag $h$-vector is defined by the relation $h_{S} = \sum_{T \subseteq S} (-1)^{|S-T|} f_{T}$. The standard way to encode the flag $h$-vector is to express it as a non-commutative polynomial in the variables ${{\bf a}}$ and ${{\bf b}}$ as follows. Define a monomial $u_S = u_1 u_2 \ldots u_n$ by $$u_i = \left\{\begin{array}{c c}
             {{\bf a}}, & i \not \in S \\
             {{\bf b}}, & i \in S,
             \end{array}\right.$$ and let $\Psi_P({{\bf a}},{{\bf b}}) = \sum_{S \subseteq [n]} h_S u_S$. The polynomial $\Psi_P({{\bf a}},{{\bf b}})$ is called the [**ab**]{}-index of $P$.

A poset $P$ is called Eulerian if for all $x \leq y$ in $P$, we have $\mu(x,y) = 