85 problems

Stanley–Stembridge conjecture. Every claw-free graph is Schur-positive.

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

For a permutation $w$, let $\mathrm{S}_{w}$ denote the Stanley symmetric function. A symmetric function is Schur positive if its expansion in the Schur-function basis has only nonn…

Fomin–Fulton–Li–Poon's conjecture. The difference

Gasharov–Stanley conjecture. The chromatic symmetric functions of all claw-free graphs are Schur positive.

Let $k$ be a positive integer and consider the product over all positive integers congruent to $1$ modulo $k$: … A symmetric function is Schur positive if its expansion in the Schu…

A finite simple graph $G$ is Schur-positive if its chromatic symmetric function $X_G$ is Schur-positive, meaning that all coefficients in its Schur-function expansion are nonnegati…

Gasharov's claw-free Schur-positivity conjecture. The chromatic symmetric function $X_G$ of every claw-free graph $G$ is Schur-positive.

Let $\mu$ be a partition, and let $\Pi$ range over bi-brick permutations. Associate to each $\Pi$ a partition statistic $\mu(\Pi)$, a composition statistic $\alpha(\Pi)$, and a non…

Higher-order Macdonald positivity conjecture. For every partition $\mu$, the multivariate $q,t$-Kostka number $\tilde{K}^{(q,t)}_{\mu;\lambda^1,\dots,\lambda^r}$ is a polynomial in…

Reutenauer's conjecture. For $n\geq2$, the symmetric polynomial $-q_{(n)}$ is Schur positive. The paper states that this conjecture was later established by W.M.Doran IV, so it is…

Positivity conjecture. The coefficients in these Schur-basis expansions are polynomials with nonnegative integer coefficients.

Let $\mathcal G^{(n)}_\lambda(q,x)$ be the symmetric functions arising from ribbon tableaux and Fock-space representations of the quantum affine algebra…

Let $\lambda/\mu$ and $\nu/\rho$ be skew shapes such that $\lambda+\nu$ and $\mu+\rho$ have only even parts. For a partition with all even parts, write…

Tilde-operation conjecture for skew Schur functions. For all $\mu/\alpha$ and $\nu/\beta$, if $(\mu/\alpha,\nu/\beta)^\sim=(\lambda/\sigma,\rho/\tau)$, then

Let $\lambda/d/\mu$ be a cylindric skew shape, and suppose inductively that … where $c_{\nu,e}\geq 0$ and $e$ is a non-negative integer; for nonzero $s_{\nu/e/\emptyset}(x)$, requi…

Let $\lambda/d/\mu$ be a cylindric skew shape that is a subposet of $\mathfrak{C}_{k,n-k}$. For variables $x=(x_1,x_2,\ldots)$, call $s_{\lambda/d/\mu}(x)$ cylindric Schur-positive…

Fomin–Fulton–Li–Poon conjecture for skew partitions. For any skew partitions $\mu/\alpha$ and $\nu/\beta$, if

Let $a\leq b<c\leq d$ be integers with $n=ad=bc$. For complete homogeneous symmetric functions, write $h_n\circ h_k$ for their plethysm, and let $\{s_\mu\}_{\mu\vdash n}$ be the Sc…

Let $n$ be a positive integer, and let $mu$ and $lambda$ be partitions of $n$. Let $m_\mu$ be the monomial symmetric function, $s_\lambda$ the Schur function, and $\nabla$ the Berg…

Let $\mathfrak S(\mathbf z)$ be the normalized Schur generating function … and, for a sequence $\alpha$, let $\partial^\alpha=\prod_{r\geq1}(\partial/\partial z_r)^{\alpha_r}$. Let…

Let $mathcal P$ be the set of partitions, and let $a,b,c,d:\mathcal P^\ell\to\mathbb R_{\geq0}$ be finitely supported functions satisfying … Write $s_\lambda$ for the Schur functio…

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…

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t^{-1}]$ be the modified Hall–Littlewood polynomial specialized at $q=0$, let $\nabla$ be the nabla operator, an…

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t]$ be the modified Hall–Littlewood polynomial obtained by specializing the modified Macdonald polynomial at…