e-log-concavity conjecture for chromatic quasisymmetric functions

At least 1 year old · documented by 2024

Let $G=([n],E)$ be a natural unit interval graph, and write

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

A polynomial $c_\lambda(q)$ with coefficients $a_\ell$ is $e$ -log-concave when its coefficients satisfy $a_\ell^2\geq a_{\ell+1}a_{\ell-1}$ for every relevant index $\ell$ .

$e$ -log-concavity conjecture. Let $G$ be a natural unit interval graph. Then $X_G(\bm x;q)$ is $e$ -log-concave.

This is proposed as a strengthening of $e$ -unimodality. The source reports computer verification for every natural unit interval graph with at most $10$ vertices, but no general proof.

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

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).

References

Primary source

Bruce E. Sagan and Foster Tom, “Chromatic symmetric functions and change of basis”, arXiv:2407.06155 (2024).

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