2003/11/24 by John Shareshian, Shareshian, John, David L. Wright +4 · 1 citation
Chemistry · Mathematics · #06A07 #06A11 #11B68 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Molecular spectroscopy and chirality #math.CO #msc:06A07 #msc:06A11 #msc:11B68
paper · pdf · doi:10.48550/arxiv.math/0311426
Latex 23 pages
arxiv created 2003/11/24 · openalex publication_date 2003/11/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.