2022/10/07 by Logan Crew, Crew, Logan, Yongxing Zhang +1
Medicine · #05C15 #05E05 #Cholesterol and Lipid Metabolism #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2210.03803
openalex publication_date 2022/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A well-known result of Stanley's shows that given a graph G with chromatic symmetric function expanded into the basis of elementary symmetric functions as XG = ∑ cλeλ, the sum of the coefficients cλ for λ with λ1' = k (equivalently those λ with exactly k parts) is equal to the number of acyclic orientations of G with exactly k sinks. However, more is known. The sink sequence of an acyclic orientation of G is a tuple (s1,…,sk) such that s1 is the number of sinks of the orientation, and recursively each si with i > 1 is the number of sinks remaining after deleting the sinks contributing to s1,…,si-1. Equivalently, the sink sequence gives the number of vertices at each level of the poset induced by the acyclic orientation. A lesser-known follow-up result of Stanley's determines certain cases in which we can find a sum of e-basis coefficients that gives the number of acyclic orientations of G with a given partial sink sequence. Of interest in its own right, this result also admits as a corollary a simple proof of the e-positivity of XG when the stability number of G is 2. In this paper, we prove a vertex-weighted generalization of this follow-up result, and conjecture a stronger version that admits a similar combinatorial interpretation for a much larger set of e-coefficient sums of chromatic symmetric functions. In particular, the conjectured formula would give a combinatorial interpretation for the sum of the coefficients cλ with prescribed values of λ1' and λ2' for any unweighted claw-free graph (not necessarily an incomparability graph, as in the setting of the Stanley-Stembridge conjecture).