599 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…

For $n=2$, write the Chern class in the Schur basis as … and expand each coefficient in the binomial basis: … A polynomial is binomially positive when all its binomial-basis coeffi…

Prove the six affine Jacobi–Trudi identities conjectured by Ole Warnaar for the relevant specialisations of multiparameter Hall–Littlewood polynomials of types $\mathrm{B}_n$,…

For every positive integer $k$ and every $k$-bounded partition $\lambda$, the $K$-$k$-Schur function $\mathsf{KSchur}^{(k)}_{\lambda}$ is a nonnegative integer linear combination o…

For every partition $\lambda$, specialize the Koornwinder moment indexed by $\lambda$ to $t=q$, normalize it as in Rains' formulation, and write the resulting rational function in…

The conjecture concerns a determinantal expression for a related antisymmetrizer arising in the study of symmetric functions and alternating-sign-matrix enumeration. The supplied s…

For every integer $n\ge 1$ and every pair of partitions $\mu\subseteq\lambda$, let $s_{\lambda/\mu}(x_1,\ldots,x_n)=\sum_{\alpha\in\mathbb{Z}_{\ge 0}^n}c_\alpha x^\alpha$ be the sk…

Let $\lambda$, $\mu$, $\lambda'$, and $\mu'$ be partitions in $\mathbb{Z}^n$. For vectors $x,y\in\mathbb{Z}^n$, define … A symmetric polynomial is Schur nonnegative if it is a nonn…

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…

Let $\mathcal P$ denote the set of partitions, let $\widetilde{G}_{\lambda}$ be the modified stable Grothendieck symmetric function, and let $\leqslant_{\operatorname{g}}$ denote G…

Fix partitions $\mu$ and $\nu$. For each family $\mathcal F$, let … and let $M_{\mu\nu}$ be the set of partitions $\lambda\supseteq\mu,\nu$ for which there exists a Molev tableau o…

Stanley--Gasharov conjecture (1998). Every claw-free graph is Schur-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 $K_{1,3}$…

Schur positivity conjecture. The coefficients $a^\nu_{\lambda\mu}(t)$ are nonnegative integer polynomials in $t$.

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…

For each positive integer $n$, let $H_n(\mathbf{x};q)$ be the Hall–Littlewood polynomial used in the paper, and for $0<a\leq b$ define … Let $f_{a,b}=h_b[h_a]-h_a[h_b]$. The q-Foul…

Let $n$ be a positive integer, let $\lambda\vdash n$ be a partition, and let $\phi^\lambda$ denote the corresponding monomial trace. For an $n\times n$ matrix $x$, write…

Let $\lambda$ be a partition and let $\theta^{(\alpha)}_\mu(\lambda)$ be the Jack-character coefficient defined by the paper, with $b:=\alpha-1$. Let $m_1(\mu)$ denote the number o…

Macdonald superpolynomial positivity conjecture. For all superpartitions $\Lambda$ and $\Om$, the coefficients satisfy

Let $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define the coefficient $f^\pi_{\mu,\nu}(\alpha)$ by … where $e_\pi$ denotes the elementary symmetric function. Monomial positiv…

Let $H$ be a type A Hessenberg space. Let $\mathfrak{ch}(\mathrm{L}_{H})$ be the graded character of the associated left representation, and let $\operatorname{Frob}$ denote the Fr…

Let $A$) be a totally nonnegative $n\times n$ matrix, meaning that all its minors are nonnegative, and let $\lambda\vdash n$ be a partition. For a class function $\phi^{\lambda}$ o…

Let $v=12\dots k$ be the increasing permutation pattern, let $N_{12\dots k}$ count occurrences of this pattern in permutations, and let $\operatorname{ch}_n(RN_{12\dots k})$ denote…

For a symmetric function $f$, let $f(x_1,\ldots,x_k)$ denote its specialization obtained by setting $x_m=0$ for all $m\geq k+1$. A polynomial…

Let $R=R(\eta,\gamma)$ be a dominant sequence of rectangles, and let $R'$ be obtained from $R$ by replacing a subsequence … by … where $|\alpha|=|\beta|$ and $\alpha^+$ dominates…