22 problems

Let $\mathbf{a}$ be an area sequence and $\mathbf{s}$ a set of strict edges, and let $\mathrm{G}_{\mathbf{a},\mathbf{s}}(\mathbf{x};q)$ be the corresponding vertical-strip LLT poly…

Let stable atoms be the basis elements ${\mathcal A}_{\eta|\lambda}$ defined from Demazure atoms, and say that an element is stable atom positive when it is a nonnegative linear co…

Let $\mathcal P(r)$ be the polynomial space in which the stable atoms form a basis, and let stable atom positivity mean that the coefficients in this basis belong to…

Let $F_\kappa(x;t^{-1})$ denote the relevant flagged LLT-type polynomial and let $w_0\mathcal D_\mu(x)$ be the opposite Demazure character. In the single-row, equal-end-content cas…

Let ${\boldsymbol\nu},\sigma$ be any flagged LLT indexing data, and let $\mathcal G_{{\boldsymbol\nu},\sigma}[x_1,\ldots,x_l;t]$ be the associated flagged LLT polynomial. If … then…

Let ${\boldsymbol\nu}$ be a tuple of ordinary, non-ragged-right skew diagrams, and let $\sigma$ be the standard compatible permutation. Let…

Let $\bm{\lambda}/\bm{\mu}$ be a sequence of unicellular shapes. Let $G_{\bm{\lambda}/\bm{\mu}}$ be the graph associated with this sequence, and for each relevant spanning tree $T$…

Let $\bm{\nu}=(\lambda^1/\mu^1,\lambda^2/\mu^2,\lambda^3/\mu^3)$ be a triple of skew diagrams. For every triple $\{i,j,k\}=\{1,2,3\}$ with $i<j$, let…

Let $\bm\lambda$ and $\bm\mu$ be horizontal strips, and let $\Pi(\bm\lambda)$ and $\Pi(\bm\mu)$ be their weighted graphs. If … as weighted graphs, then the associated LLT polynomia…

Let $\bm\lambda$ be a horizontal strip, and let $\Pi(\bm\lambda)$ be its weighted graph with vertices labelled $v_1,v_2,\ldots$. Write $M_{j,k}$ for the relevant edge weight and…

For an indifference graph $G$, let … where the sum runs over all colorings of $G$. The polynomial is a vertical-strip LLT polynomial, and a symmetric function is $e$-positive if it…

Let $D$ be a Dyck path in the $n\times n$ lattice square $R_n$. Form $\operatorname{dinvset}(D)$ from the pairs of cars producing a dinv in the parking function with permutation…

Let $\mathbf{a}$ be an area sequence of length $n$, let $O(\mathbf{a})$ be the set of orientations associated with $\mathbf{a}$, let $P(\theta)$ be the poset obtained from the asce…

Let $\mathbf{a}$ be an area sequence, let $O(\mathbf{a})$ be the set of the relevant orientations of its graph, let $\operatorname{asc}(\theta)$ be the ascent statistic, and let…

Haiman–Hugland's conjecture. The symmetric function

Let $G$ be an oriented graph with no loops or multiple edges. Write the relevant LLT polynomial expansion as … where $\Psi_{\alpha}$ is the quasisymmetric power sum indexed by the…

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…

Let ${\boldsymbol\nu}$ determine a circular vertical-strip digraph, let $\Gamma_{\boldsymbol\nu}$ be its underlying digraph, and let $O_\ast(\Gamma_{\boldsymbol\nu})$ denote the or…

Let ${\boldsymbol\nu}$ determine a circular vertical strip digraph. Circular LLT power-sum positivity conjecture. Then $\omega \mathrm{G}_{\boldsymbol\nu}(\mathbf{x};q+1)$ is…

Let $\Gamma$ be a circular unit arc digraph, and let $H$ be a graph obtained from $\Gamma$ by marking $k$ corner edges strict. Strict-corner difference conjecture. Then … is…

Let ${\boldsymbol\nu} = (\mathbf{a},\mathbf{s})$ be a circular Dyck diagram with some strict corner edges. Circular LLT e-positivity conjecture. Then…

Noncommutative Schur positivity conjecture. For every integer partition $\lambda$ and every positive integer $k$, $\mathfrak{J}_\lambda(\boldsymbol{u})$ is $\mathbb{Z}$-monomial po…