The Shareshian-Wachs conjecture: elementary positivity of chromatic quasisymmetric functions of natural unit interval graphs

Theorem 1.1: a terminating procedure assigns to each $\sigma\in D_G^0$ (permutations whose adjacent descents are edges) a partition $\theta_G(\sigma)\vdash n$ with $X_G(x;q)=\sum_{\sigma\in D_G^0}q^{\mathrm{ginv}_G(\sigma)}e_{\theta_G(\sigma)}(x)$, where $\mathrm{ginv}_G$ counts inversions supported on edges. Hence every elementary coefficient lies in $\mathbb N[q]$ and counts permutations; at $q=1$ this reproves the Stanley-Stembridge conjecture (Hikita; Griffin-Mellit-Romero-Weigl-Wen). Not shown: the unimodality part of the Shareshian-Wachs conjecture, an efficient description of $\theta_G$ (the procedure may be slow), or $e$-positivity outside the natural unit interval class.

Theorem 1.1: a terminating procedure assigns to each $\sigma\in D_G^0$ (permutations whose adjacent descents are edges) a partition $\theta_G(\sigma)\vdash n$ with $X_G(x;q)=\sum_{\sigma\in D_G^0}q^{\mathrm{ginv}_G(\sigma)}e_{\theta_G(\sigma)}(x)$, where $\mathrm{ginv}_G$ counts inversions supported on edges. Hence every elementary coefficient lies in $\mathbb N[q]$ and counts permutations; at $q=1$ this reproves the Stanley-Stembridge conjecture (Hikita; Griffin-Mellit-Romero-Weigl-Wen). Not shown: the unimodality part of the Shareshian-Wachs conjecture, an efficient description of $\theta_G$ (the procedure may be slow), or $e$-positivity outside the natural unit interval class.