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

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

2003-11-24

arxiv.org/abs/math/0311426v1

Mathematics

Combinatorics

Latex 23 pages

Scientific paper

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.

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