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

TL;DR

This paper introduces a novel recursive approach to compute order and Eulerian polynomials of labeled posets, connecting these to adjacency matrices, Bernoulli numbers, and generating new invariants.

Contribution

It provides formulas, recursion relations, and a recursive algorithm for invariants of labeled posets, unifying various polynomial invariants and extending existing quasi-symmetric function frameworks.

Findings

Derived formulas for order and Eulerian polynomials using the ω-graph adjacency matrix

Established recursion formulas and applications to Bernoulli numbers and polynomials

Developed a recursive algorithm for constructing invariants of labeled posets

Abstract

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.