49 problems

Chromatic symmetric homology distinction conjecture. Each such pair is distinguished by chromatic symmetric homology.

Let $G$ be a simple graph, and let $X_G({\bf x})$ denote its chromatic symmetric function. A graph is claw-free if it has no induced subgraph isomorphic to the claw graph $K_{1,3}$…

Let $G$ and $H$ be trees, and let $\chi_G(x)$ denote the chromatic symmetric function of $G$. Tree isomorphism conjecture. If $G$ and $H$ are non-isomorphic trees, then … The conje…

Let a spider be a tree with at most one vertex of degree greater than two, and let $X_T^T$ denote the self-chromatic symmetric function of a spider $T$. Spider self-CSF conjecture.…

Let $\textbf m$ and $\textbf n$ denote chains indexed by the positive integers $m$ and $n$, and let $\textbf m\times\textbf n$ be their product lattice. A lattice is Schur positive…

Let $B_n$ denote the Boolean lattice of subsets of an $n$-element set, and let a poset be nice when its incomparability graph has the nice property. Boolean-lattice niceness conjec…

Let $\mathbf{m}\in\mathbb{E}_n$, let $P=P_{\mathbf{m}}$ be the associated natural unit interval order, and let $\lambda\vdash n$. Let $\operatorname{strongST}_P(\lambda)$ be the st…

Let $P$ be a $(\mathbf{3}+\mathbf{1})$-free poset, let $\lambda$ be a partition, and let $c_\lambda^P$ be the corresponding elementary-basis coefficient of…

Let $P$) be a $(\mathbf{3}+\mathbf{1})$-free poset, let $\lambda$ be a partition, and let $c_\lambda^P$ denote the coefficient of $e_\lambda(\mathbf{x})$ in…

Let $G_1$ and $G_2$ be connected unicyclic graphs, where either each has a 3-cycle and an odd number of vertices, or each has a $c$-cycle for some $c\geq4$. Let $\mathbf{X}_G$ deno…

Let $T$ be a tree, let $\Delta(T)$ denote its maximum degree, and say that $T$ is $e$-positive when its chromatic symmetric function $X_T$ has a nonnegative expansion in the elemen…

For a proper smooth variety $\mathscr X$ equipped with a $G[\![z]\!]$-action and a $G[\![z]\!]$-equivariant map … assume that … where…

Wang's conjecture. All theta graphs $\theta_{abc}$ are $e$-positive.

For positive integers $m$ and $n$, let $\mathbf m\times\mathbf n$ be the product of two chains, and let $\operatorname{inc}(\mathbf m\times\mathbf n)$ denote its incomparability gr…

Hat-chain e-positivity conjecture. Every hat-chain is $e$-positive.

Twin-free reduction conjecture. If $P$ is $(3+1)$-free and twin-free, then $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive.

Foley–Hoàng–Merkel's twinning conjecture. If $P$ is $(3+1)$-free and $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive, then $X_{\operatorname{Inc}(P_v)}(\mathbf{x})$ is $e$-…

For positive integers $m,n$, let $S(a,b,c)$ denote the spider formed from three paths of lengths $a,b,c$ with a common end. A graph is $e$-positive when its chromatic symmetric fun…

For a partition $ambda=ambda_1\dotsmambda_d$ of $n-1$ with $d\ge 3$, the spider $S(\lambda)$ is the $n$-vertex tree consisting of paths of lengths $\lambda_1,\dots,\lambda_d$ with…

For any integers $a\ge b\ge c\ge 1$, let $\theta_{abc}$ be the theta graph formed from two distinct vertices joined by three pairwise disjoint paths of lengths $a$, $b$, and $c$. A…

Stanley–Stembridge conjecture. If $P$ is a ${\bf 3}+{\bf 1}$-avoiding partial order, then $X_G$ is $e$-positive.

Crew's conjecture. For any tree $T$, the generalized degree sequence of $T$ is determined by $X_T$.

Let $(G,\omega)$ be a set-weighted graph with $n$ vertices and total weight $d$, and write … Let $\mu\leq d$ be an integer, viewed as a one-part partition, that is $s$-allowable in…

Rooted $(3+1)$-free $e$-positivity conjecture. For every $r\in P$, if $P-r$ is $(\mathbf{3}+\mathbf{1})$-free, then

Finite principal-specialization conjecture. For all such trees,