e-log-concavity conjecture for chromatic quasisymmetric functions

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. e-log-concavity conjecture for chromatic quasisymmetric functions

   Let $G$ be a natural unit interval graph, and write

   $$X_G(\bm x;q)=\sum_\mu c_\mu(q)e_\mu.$$

   A polynomial $a(q)=\sum_{k=i}^j a_kq^k\in\mathbb N[q]$ is log-concave when $a_k^2\geq a_{k-1}a_{k+1}$ for every $k$. The e-log-concavity conjecture. The function $X_G(\bm x;q)$ is $e$-log-concave: for every partition $\mu$, the coefficient $c_\mu(q)$ of $e_\mu$ is a log-concave polynomial. The author reports computer verification for every natural unit interval graph with at most $10$ vertices. Whether this holds in general remains open.

   source: Foster Tom, “A signed e-expansion of the chromatic quasisymmetric function”, arXiv:2311.08020 (2024).