  In this paper, we first give formulas for the order polynomial $\Omega (\Pw;
t)$ and the Eulerian polynomial $e(\Pw; \lambda)$ of a finite labeled poset
$(P, \omega)$ using the adjacency matrix of what we call the $\omega$-graph of
$(P, \omega)$. We then derive various recursion formulas for $\Omega (\Pw; t)$
and $e(\Pw; \lambda)$ and discuss some applications of these formulas to
Bernoulli numbers and Bernoulli polynomials. Finally, we give a recursive
algorithm using a single linear operator on a vector space. This algorithm
provides a uniform method to construct a family of new invariants for labeled
posets $(\Pw)$, which includes the order polynomial $\Omega (\Pw; t)$ and the
invariant $\tilde e(\Pw; \lambda) =\frac {e(\Pw;
\lambda)}{(1-\lambda)^{|P|+1}}$. The well-known quasi-symmetric function
invariant of labeled posets and a further generalization of our construction
are also discussed.
