What about the Stanley–Stembridge conjecture on (3+1)-free posets? Given a poset $P$ with vertex set $V = \{1,2,\ldots,n\}$, define the symmetric function $X_P$ in countably many indeterminates $x_1, x_2, \ldots$ by $$X_P := \sum_{\kappa:V\to\mathbb{N}} x_{\kappa(1)}x_{\kappa(2)}\cdots x_{\kappa(n)}$$ where the sum is over all maps $\kappa:V \to\mathbb{N}$ such that the pre-image $\kappa^{-1}(i)$ of every $i\in\mathbb{N}$ is a totally ordered subset of $P$. Then the conjecture is that if $P$ is (3+1)-free (i.e., that $P$ contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain) then the expansion of $X_P$ in terms of elementary symmetric functions has nonnegative coefficients.

This conjecture grew out of certain positivity conjectures about immanants. Guay-Paquet has reduced the conjecture to the case of unit interval orders (i.e., posets that are both (3+1)-free and (2+2)-free), and the unit interval order case has a graded generalization (due to Shareshian and Wachs), which has close connections with representation theory and Hessenberg varieties.