47 problems

Let $\mathcal{I}^*_{\alpha}$ denote the dual immaculate quasisymmetric function indexed by a composition $\alpha$. Zabrocki's conjecture. The dual immaculate quasisymmetric functio…

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 $(P,\omega)$ 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…

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 $\mathcal{K}_n$ be the cone of symmetric functions in the fundamental quasisymmetric-function basis whose coefficients are nonnegative, and let $\beta\sim\gamma$ denote the equ…

Let $S\subseteq S_n$ be a symmetric set of permutations, meaning that … is symmetric, and suppose that $S\not\subseteq\{\iota_n,\delta_n\}$, where $\iota_n:=12\dots n$ and…

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 $\PCp$ denote the set of peak compositions, and for each peak composition $\alpha$ let $K_\alpha$ be the corresponding peak quasisymmetric function. The quasisymmetric Schur…

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 ${\underline\lambda}=(\lambda^+,\lambda^-)\vdash n$ be a bipartition, let $\psi^{\underline\lambda}$ be the type $B$ higher Lie character indexed by ${\underline\lambda}$, and…

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…

Grinberg's conjecture. The statistics $\operatorname{Lpk}$, $\operatorname{lpk}$, $\operatorname{Rpk}$, $\operatorname{rpk}$, $\operatorname{Val}$, and $\operatorname{val}$ are $M$…

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 $f(t)cin 1+t\mathbb{Z}[[t]]$. An upho poset is a finite-type poset isomorphic to each of its principal order filters, and its Ehrenborg quasisymmetric function records the rank…

Conjecture on larger pattern sets. Based on computer experiments, the behavior changes when

Let $\Pi\subseteq {\mathfrak S}_3$ be a pattern set in the case under discussion, and let $Q_n(\Pi)$ denote the associated quasisymmetric function. A nice Schur expansion conjectur…

Let the Leibniz-Hopf algebra be the Hopf algebra described in the source, and let duality be taken for Hopf algebras over the integers. Ditters' conjecture. The Leibniz-Hopf algebr…

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…