24 problems

Let $G$ be a graph, and let $T\chi_G^o(q)$ denote its total oriented chromatic quasisymmetric function. Say that cycles of $G$ are edge-disjoint when no two cycles share an edge. S…

Let $k$ be a positive integer, let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, and let $G_h$ be the graph associated with $h$. For a proper coloring $\kappa$ of $G_h$, w…

Let $G$ and $H$ be trees equipped with orientations $\gamma$. For an orientation $\gamma$, let $\chi_G^\gamma(x;q)$ denote the oriented chromatic quasisymmetric function obtained b…

Let $G=(V,E)$ be a unit interval graph, with its vertices in the natural order. Define its chromatic quasisymmetric function by … where the sum is over proper colorings and…

Let $a,b\geq2$ be positive integers, let $\gamma=(a,2,2,\ldots,2,b)$, and let $K_\gamma$ be the corresponding K-chain with associated poset $P$. K-chain powerful-tableau formula co…

Let $P$ be a natural unit interval order with $n$ elements, let $\lambda\vdash n$, and let $\operatorname{strongST}_P(\lambda)$ be the set of strong standard $P$-tableaux of shape…

Ellzey's conjecture. If $\vec G$ is a proper circular arc digraph, then $X_{\vec G}(\bm x;q)$ is $e$-positive.

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

Let $G=([n],E)$ be a Dyck graph. For a composition $\alpha\vDash n$, let $\lambda(\alpha)$ be the partition obtained by rearranging its parts, and let $\mathcal{N}_{G,\alpha}$ be t…

Let $G=([n],E)$ be an interval graph. For a composition $\alpha\vDash n$, let $\Psi_\alpha$ be the type 1 power sum quasisymmetric function, let $z_\alpha=z_{\lambda(\alpha)}$, and…

Let $\mathbf{m}\colon[n]\to[n]$ be a Hessenberg function, let $G_\mathbf{m}$ be its indifference graph, and let $g_k(G_\mathbf{m};x,q)$ be the symmetric functions defined in the pa…

The e-positivity conjecture. For each $\gamma \in \mathcal{I}\mathcal{G}$, $X_{\gamma}(\mathbf{x};t)$ is $e$-positive; equivalently, $a_{\lambda}(t)\in\mathbb{Z}_{\geq 0}[t]$ for e…

Shareshian–Wachs conjecture. If $G$ is the incomparability graph of a natural unit interval order, then $X_G(\mathbf{x},q)$ is $e$-positive.

Let $P$ be a natural unit interval order on $[n]$, let $\mu\in\mathbb{N}^n$, and let $\mathcal{H}(P,\mu)/\sim$ be the set of equivalence classes of heaps. For an equivalence class…

Nadeau–Tewari sign conjecture. For fixed $j$, $a_j^{m,n}(\lambda,q)$ is a Laurent polynomial in $q$ whose coefficients are integers of the same sign.

Let $P(\mathbf{m})$ be the natural unit interval order associated with a sequence $\mathbf{m}=(m_1,m_2,m_3,n,\ldots,n)$, and let $G$ be its incomparability graph. Let…

Let $P(\mathbf{m})$ be the natural unit interval order associated with a sequence $\mathbf{m}=(2,m_2,m_3,n,\ldots,n)$, and let $G$ be its incomparability graph. Let…

Circular indifference digraph conjecture. The palindromic polynomial $X_{\overrightarrow{G}}({\bf x},t)$ is $e$-positive and $e$-unimodal: $a_j({\bf x})$ is $e$-positive for every…

Let $\mathbf{a}$ be a circular area sequence, and let $\mathrm{G}_{\mathbf{a}}(\mathbf{x};q+1)$ and $\mathrm{X}_{\mathbf{a}}(\mathbf{x};q)$ denote the corresponding LLT polynomial…

Circular chromatic coefficient conjecture. The coefficients $c_\lambda(q)$ are palindromic and unimodal polynomials with non-negative integer coefficients. This extends and refines…

Let $\mathbf{a}$ be a circular area sequence, and let $\mathrm{X}_{\mathbf{a}}(\mathbf{x};q)$ be the corresponding chromatic symmetric function. Circular chromatic e-positivity con…

Let $\overrightarrow{G}$ be a proper circular arc digraph, meaning a digraph with no induced subdigraph isomorphic to either $\overrightarrow{K_{12}}$ or $\overrightarrow{K_{21}}$.…

Let ${\bf m}$ be a Hessenberg vector, and let $P({\bf m})$ be the natural unit interval order on $[n]$ defined by $i<_{P({\bf m})}j$ if $j>m_i$. Let ${\mathcal H}({\bf m})$ be the…

Let $P$ be a natural unit interval order on $[n]$ with incomparability graph $G$. For a partition $\mu=(\mu_1\geq\cdots\geq\mu_l)\vdash n$, let $z_\mu$ be the standard symmetric-gr…