97 problems

Stanley–Stembridge conjecture. Every claw-free graph is Schur-positive.

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…

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 $G=(V,E)$ be a finite tree. Its generalized degree polynomial (GDP) is … where $\operatorname{ext}(A)$ is the number of edges of $E$ with one endpoint in $A$, and…

Let $n$ be the order of a tree, and let $\Psi_n(x,y)$ be expanded in the homogeneous symmetric-function basis as … For a partition $\mu\vdash n$, let $\epsilon(\mu)$ be the number…

Let $X_G$ be the chromatic symmetric function of a graph $G$, and let $X_G(x_1,\ldots,x_k)$ denote its specialization to $k$ variables. A polynomial is SNP (has a saturated Newton…

A finite simple graph $G$ is Schur-positive if its chromatic symmetric function $X_G$ is Schur-positive, meaning that all coefficients in its Schur-function expansion are nonnegati…

Gasharov's claw-free Schur-positivity conjecture. The chromatic symmetric function $X_G$ of every claw-free graph $G$ is Schur-positive.

Let $T_1$ and $T_2$ be non-isomorphic finite trees. Their chromatic symmetric functions are denoted by $X_{T_1}$ and $X_{T_2}$, respectively. Stanley's conjecture. Non-isomorphic t…

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 $e$-…

Stanley's tree problem. For two trees $T$ and $F$, $X_T=X_F$ if and only if $T\cong F$.

Stanley’s tree isomorphism conjecture. The chromatic symmetric function distinguishes trees: if $T$ and $T'$ are trees and $X_T=X_{T'}$, then $T$ and $T'$ are isomorphic.

Stanley's e-positivity conjecture. If $G$ is a claw-free incomparability graph, then $X_G$ is e-positive; equivalently,

Let $n$ be the order of a tree, and let $\Psi_n(x,y)$ be expanded in the homogeneous symmetric-function basis as … where $z_\mu$ is the standard symmetric-group factor associated w…

The (3+1)-free conjecture. For every $(3+1)$-free graph $G$, the chromatic symmetric function $X_G$ is $e$-positive.

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 $G$ be a graph and let $H=S_{n+1}$ be the $(n+1)$-vertex star graph, consisting of one central vertex and $n$ other vertices connected only to the central vertex. Let $X_G^H$ b…

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 $G$ be a finite tree, and let $X_G^G$ denote its self-chromatic symmetric function (self-CSF). The self-CSF conjecture for trees. The self-CSF distinguishes all trees from each…

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 $G=(V,E)$ be a graph, let $X_G(x_1,x_2,\ldots)$ be its chromatic symmetric function, and write $X_G(q,q^2,\ldots,q^n,0,0,\ldots)$ for its principal specialization. Loehr–Warrin…

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…