A New Approach to Order Polynomials of Labeled Posets and Their Generalizations

In this paper, we first give formulas for the order polynomial $\Omega (\Pw; t)$andtheEulerianpolynomial$e(\Pw; \lambda)$ofafinitelabeledposet$(P, \omega)$usingtheadjacencymatrixofwhatwecallthe$\omega$-graphof$(P, \omega)$\mathrm{.}Wethenderivevariousrecursionformulasfor$\Omega (\Pw; t)$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 $(\text{Pw})$, which includes the order polynomial $\mathrm{\Omega }(\text{Pw};t)$ 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.