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

Abstract

In this paper, we first give formulas for the order polynomial $Ω(\Pw; t)$ and the Eulerian polynomial $e(\Pw; λ)$ of a finite labeled poset $(P, ω)$ using the adjacency matrix of what we call the $ω$-graph of $(P, ω)$. We then derive various recursion formulas for $Ω(\Pw; t)$ and $e(\Pw; λ)$ 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 $Ω(\Pw; t)$ and the invariant $\tilde e(\Pw; λ) =\frac {e(\Pw; λ)}{(1-λ)^{|P|+1}}$. The well-known quasi-symmetric function invariant of labeled posets and a further generalization of our construction are also discussed.