185 problems

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…

A primitive distance-regular graph is a distance-regular graph with no nontrivial equivalence relation coming from an imprimitive association scheme. Bannai–Ito conjecture. Any pri…

Let $L$ be a Latin square of order $n$, and define its sign by the product of the signs of its row and column permutations. Let $E_n$ and $O_n$ denote the numbers of even and odd L…

Let $G$ be a finite group and let an $S$-ring over $G$ be a subring of $\mathbb{Z}G$ that is a free $\mathbb{Z}$-module spanned by a partition of $G$ closed under taking inverses a…

Mulmuley's conjecture. These constants belong to $\#\mathsf{P}$.

Let Schubert polynomials be the polynomial family associated with permutations. Kohnert conjectured that each Schubert polynomial admits a generating-polynomial interpretation in t…

For each permutation $\pi$, let $K_{\pi}$ be the universal Kerov character polynomial defined by … For a cycle of length $k$, write $K_k$ for the corresponding polynomial. Kerov's…

Let $\mathbb{F}$ be a field, let $M$ be a nonsingular matrix over $\mathbb{F}$, and let $x$ be a vector over $\mathbb{F}$. Alon–Jaeger–Tarsi conjecture. For any field $\mathbb{F}$…

Godsil's conjecture. If $G$ is a connected graph which is a colour class in an association scheme, then

Diameter-boundedness conjecture. The diameter of $\Gamma$ is bounded by a function in $b$.

Diaconis–Eriksson conjecture. The ideals of the Birkhoff polytopes are generated in degree $3$.

Stahl's conjecture. Every genus polynomial is real-rooted.

Let $m \geq 2$ be a fixed integer. A geometric distance-regular graph has diameter $D \geq 3$ and intersection number $c_2 \geq 2$. Koolen–Bang's classification conjecture. Any geo…

Bang–Koolen's classification conjecture. Any such graph is a Johnson graph, a Hamming graph, a Grassmann graph, a bilinear forms graph, or has a number of vertices bounded by a fun…

Let $r\geq 2$, and let $\mathcal P\subset\mathbb{C}^d$ be a set of $n$ points. Write $\mathcal{L}_r(\mathcal P)$ for the set of lines incident to at least $r$ points of…

Let $\mathrm{DH}_n$ be the space of diagonal harmonics, and let $\mathcal F(\mathrm{DH}_n)$ denote its Frobenius image. Let $e_n$ be the $n$th elementary symmetric function, … and…

Lee's tridiagonal relations conjecture. If $\Phi$ is circular, then there exist scalars $\beta,\gamma,\gamma^*,\varrho,\varrho^*$ such that

Let $G_{\lambda}$ denote the stable Grothendieck polynomial indexed by a partition $\lambda$, and let $G_{(n)}$ and $G_{1^n}$ denote the elements indexed by the one-row and one-col…

Brouwer's conjecture. The vertex-connectivity of $G$ equals its degree:

Convergence conjecture. This rewriting system is convergent, meaning that it is terminating and confluent.

Let a strongly-regular graph have Latin square or negative Latin square parameters, and suppose that its subconstituents are strongly regular. Cameron's conjecture. These are the o…

Let a polynomial have saturated Newton polytope (SNP) if its support is M-convex and its Newton polytope contains no lattice points other than those in its support. Monical–Tokcan–…

Grothendieck polynomials $\mathfrak G_w$ are multivariate polynomials associated to permutations $w\in S_n$. Their homogenization is obtained by introducing an additional homogeniz…

A unit circulant digraph of order $n$ is a Cayley digraph ${\rm Cay}({\mathbb Z}_n,S)$, where $S\subseteq{\mathbb Z}_n^*$ is the set of units in ${\mathbb Z}_n$. Toida's conjecture…

Let $U$ be the unipotent radical of a Borel subgroup in a semisimple, simply connected algebraic group defined and split over $\mathbb{R}$. The totally nonnegative part of the link…