30 problems

Let $PC_n$ be the poset of unordered pairs $\{\alpha,\beta\}$ of compositions satisfying $|\alpha|+|\beta|=n$, ordered by … For an unordered pair $\{\alpha,\beta\}$, define…

Let $cmathfrak{S}_{w}(x_1,\cdots,x_n)$ be a Schubert polynomial, and let $clangle cmathfrak{S}_{w}crangle_n^q$ denote its $q$-divided symmetrization. A symmetric polynomial is Hall–…

Let $B_n^{(1,2)}$ be the proposed monomial basis of $R_n^{(1,2)}$. For $b\in B_n^{(1,2)}$, write $\deg_x(b)$, $\deg_\theta(b)$, and $\deg_\xi(b)$ for its degrees, and let…

Let $M$ be an $\fS_n$-module, and let $X$ be a set of combinatorial objects equipped with functions $\operatorname{stat}:X\to\mathbb{N}$ and…

Let $\falR_{\fmu}$ denote the Garsia–Procesi module associated with a partition $\fmu$, and let $\fY={x^{\alpha}\mid \alpha\text{ is sub-Yamanouchi}}$ be its specified monomial bas…

Let $G=([n],E)$ be an interval graph and let $\tau\in\mathfrak{S}_n$. Let $\Phi_G(\tau)$, $\mathcal{Q}_{\Phi_G(\tau)}^{(G)}$, $\mathsf{CoInv}_G$, and $\mathsf{Inv}_G$ be the functi…

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 $\lambda$ be a partition, and let $Skeleton(\lambda)$ be the graph obtained from the crystal of tableaux $B(\lambda)$ by replacing each quasicrystal with its associated standar…

Let $P$ and $Q$ be posets with $n$ elements whose Hasse diagrams are trees. Let $\overline{K}_{P}(\mathbf{x})$ denote the strict $P$-partition enumerator, and specialize…

Let $P$ and $Q$ be finite posets whose Hasse diagrams are trees, and let $\overline{K}_{P}(\mathbf{x})$ denote the strict $P$-partition enumerator. The strict P-partition enumerato…

Let $\overrightarrow{G}$ and $\overrightarrow{H}$ be directed trees, and let $X_{\overrightarrow{G}}(\mathbf{x},t)$ denote the chromatic quasisymmetric function of a directed graph…

Let $n\geq1$, let $\mathsf{PSQ^E}(n)$ be the relevant set of square paths, and let $\delta(P)$ be the dinv reading word of $P$. Write $\mathsf{Des}(\delta(P))$ for its descent set…

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…

A weak composition is a finite sequence of nonnegative integers. For weak compositions $a$ and $b$, let $\overline{\mathfrak{P}}_a$ denote the kaon associated with $a$, and let…

A weak composition is a finite sequence of nonnegative integers. For weak compositions $a$ and $b$, let $M^a_b(\beta)$ be the coefficient of the glide polynomial…

Let $n$ be a positive integer, let $\mu$ be a partition of size $n$, let $S_n$ be the symmetric group, and let $Q_A(x)$ denote the fundamental quasisymmetric function indexed by a…

Let $\operatorname{QSym}$ be the algebra of quasisymmetric functions, let $A_-:\operatorname{QSym}\to\operatorname{QSym}$ be the derivation defined by $A_-(1)=0$ and … let…

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ be a partition whose $k$ parts are all distinct. For $\sigma\in S_k$, define … and let $\ell(\sigma)$ be the length of $\sigma$…

For $k=0,1,2$, let $\{f^{(k)}\}$ be the set of functions realized as … where $\mathcal{C}$ is a single equivalence class of $\equiv^k$. Basis conjecture. The set $\{f^{(2)}\}$ form…

Let $H=(V,E)$ be a hypertree, and let $X_H$ denote its chromatic symmetric function. The $F$-basis is the fundamental quasisymmetric-function basis. Hypertree F-positivity conjectu…

Let $(P,\omega)$ be a finite labeled poset, and let $\Gamma(P,\omega)$ be its $P$-partition enumerator, a quasisymmetric function. An isomorphism of labeled posets is a bijection p…

Permutation formulation of the rational Shuffle Conjecture. The following equation holds:

The rational Shuffle Conjecture. The following equation holds:

Let $b4$ be a diagram with $n$ cells. Write $a3am(n)$ for the set of standard Yamanouchi words of length $n$, $a3am_b4(n)$ for the subset associated with $b4$, and let $c1$ denote…

Let $A$ and $B$ be skew shapes of the same size. For each positive integer $k$, let $\mathrm{rows}_{k}(A)$ and $\mathrm{rows}_{k}(B)$ denote their row-overlap partitions, and let…