1,069 problems

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…

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

Integrality conjecture. $X$ is integral over $\Lambda$ whenever no such subset sum vanishes.

Let $K_{\lambda\mu}(q,t)$ be the coefficients obtained by expanding the modified Macdonald polynomial in a shifted Schur basis. Macdonald's positivity conjecture. The coefficients…

Let $N$ be a positive integer, and let $m,n$ be integers with $m\leqslant n$. For representations of $\operatorname{SL}_N(\mathbb{C})$, write $U\leqslant V$ when $U$ is isomorphic…

Let $\mathrm{DH}_n$ denote the space of diagonal harmonics, viewed as a bigraded $S_n$-module, and let $\operatorname{Frob}(\mathrm{DH}_n)$ denote its bigraded Frobenius characteri…

Let $\lambda,\mu,\nu$ be partitions with $|\lambda|+|\mu|=|\nu|$, and let $c^{\nu}_{\lambda\mu}$ be the Littlewood–Richardson coefficient, the multiplicity of the Schur function…

Dyck-path dimension conjecture. The dimension of $DHA_n$ is given by the number of Dyck paths in ${\cal L}_n$.

Let $n$) and $k$ satisfy $0\le k<n$, and let $\Delta'_{e_{n-k-1}}$ be the Delta operator on modified Macdonald polynomials. Let a module of super-diagonal coinvariants denote the m…

Let $w$ be a permutation, and let the Schubert polynomial corresponding to $w$ be considered for factorization into elementary symmetric polynomials. Pattern-avoidance conjecture.…

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…

Atom-positivity conjecture. The row operator applied to a key polynomial is atom-positive:

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

Let $J_{\lambda}(q,t)$ be the integral form Macdonald symmetric function and let $s_{\mu}$ be the Schur function, with $\lambda$ and $\mu$ partitions. For a non-negative integer…

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…

Degree bound conjecture. The degree bound referred to in the paper is not optimal, and

A specialization $\rho$ of the algebra of symmetric functions $\operatorname{Sym}$ is an algebra homomorphism from $\operatorname{Sym}$ to $\mathbb{C}$. It is Macdonald nonnegative…

Let $n$ be the order of a tree, and let $\Psi_n(x,y)$ be expanded in the homogeneous symmetric-function basis as … For a partition $\mu\vdash n$, let $\epsilon(\mu)$ be the number…

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 $\lambda'$ be a two-row partition, and let $\mathrm{prs}_{\lambda'}$ be the associated skew Schur partition function, defined on those partitions for which the function is defi…

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…

Rational parking function symmetry conjecture. The polynomial $PF_{m,n}(q,t)$ is symmetric in $q$ and $t$, namely

Let $c^rho{tau, u}(alpha)$ be the coefficients defined by … Here $b:=\alpha-1$ is the shifted Jack parameter. Goulden–Jackson's conjecture. The coefficients satisfy…

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…