49 problems

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 $clambdain P^+$, let … and order this set by $(clambda_1,clambda_2)succeq(cmu_1,cmu_2)$ exactly when $(clambda_1-clambda_2)(h_calpha)\leq(cmu_1-cmu_2)(h_calpha)$ for every posit…

Stanley--Gasharov conjecture (1998). Every claw-free graph is Schur-positive.

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

Let $\lambda$ and $\mu$ be partitions, and let $\lambda\cup\mu=(\nu_1,\nu_2,\nu_3,\ldots)$ be the partition obtained by rearranging all parts of $\lambda$ and $\mu$ in weakly decre…

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…

Generalized Amdeberhan–Shareshian–Stanley conjecture. For every partition $\mu$ of $kn$, the symmetric function obtained by applying the partition division map…

Butler's conjecture. The symmetric function $\operatorname{I}_{\lambda,\mu}[X;q,t]$ is Schur positive.

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…

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 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 $n$ 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 $\lambda$ 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…

Let $\lambda$ be a partition and let $\alpha$ be an integral vector. Arithmetic-progression Schur log-concavity conjecture. The sequence … is strongly Schur log-concave for every s…

Let $\rho,\mu,\nu,\delta$ be partitions, and let $\chi_\lambda$ denote the corresponding $\operatorname{SL}_n$ character. Suppose … and that $\mu$ and $\nu$ belong to the minimal a…

Let $\lambda, \beta$ be partitions and let $\alpha$ be an integral vector. Suppose $ell(\lambda) gg ell(\alpha)$ and $\lambda_{ell(\lambda)} gg \beta_1$. Growing-shape strong Schur log-con…

For positive integers $m$ and $n$, let $\mathbf m\times\mathbf n$ be the product of two chains, and let $\operatorname{inc}(\mathbf m\times\mathbf n)$ denote its incomparability gr…

Let $\Lambda$ be the ring of symmetric functions, let $m_{\mu}$ and $s_{\lambda}$ denote the monomial and Schur symmetric functions indexed by partitions $\mu$ and $\lambda$, respe…

A curve $C$ passing through lattice points $(0,0)={\bm p}_0,{\bm p}_1,\ldots,{\bm p}_k=(m,n)$ is ${\mathbb Z}$-convex if the set of lattice points strictly above $C$, together with…

Equidistribution conjecture. For all $n\in\mathbb{N}$,

Stembridge's conjecture. Monomial immanants of Jacobi–Trudi matrices are Schur-positive.