19 problems

Let $A$ be a $d \times n$ matrix with $n-d=3$, and let $\mathcal{G}_A$ denote its Gröbner fan. Sturmfels–Thomas conjecture. Every maximal cone in $\mathcal{G}_A$ has three or four…

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…

Let $M$ be a matroid and let $I_M$ be its toric ideal. A quadratic binomial in $I_M$ is a binomial homogeneous of degree two in the base variables. The quadratic-generation conject…

Let $M$ be a matroid on the ground set $\{1,2,\ldots,n\}$. Fix a field $k$, let $S_M=k[y_B:B\text{ a base of }M]$, and let $I_M$ be the kernel of the homomorphism…

Let $S$ be a polynomial ring, let $\succ$ be a term order on $S$ other than the graded reverse lexicographic order, and let $k\geq 2$ be the smallest degree on which $\succ$ and th…

Upper-bound conjecture. Then

Existence range conjecture. There exists a bi-Cohen–Macaulay simplicial complex $\Delta$ that is not a cone if and only if

Choffrut's conjecture. For $n=4$, the grammic congruence is generated by the Knuth relations together with the pairs

For $n\geq 1$, define the reversal permutation … Let $r_n^{\pi_n}$ be the corresponding element of $\mathfrak{U}_{\mathsf{DY}}^1$, and let $p_{n,m}$ denote the coefficient of…

Let $c_{n,m}$ denote the coefficient introduced in Proposition, and let $r_n^\mathsf{id}$ and $r_m^\mathsf{id}$ be the corresponding elements of the universal Drinfeld–Yetter algeb…

Let $\lambda$ be a partition, let $\mu$ and $\kappa$ define the 2-colour separation $\mathcal{C}^{B_{\lambda}}_{|\mu|,|\kappa|}$, and let…

Let $\bbeta=(1,2,\ldots,\ell(\lambda),|\mu|,|\kappa|)$ define the $1\times(\ell(\lambda)+2)$ integer matrix … Here $\lambda$ is a partition with $\ell(\lambda)\geq 5$, $\mu$ and…

Thinness conjecture. ${\mathbb P}$ is thin if and only if

Assaf–Searles' conjecture. Given a southwest diagram $D$, the Kohnert polynomial ${\mathfrak{K}}_D$ expands non-negatively into Demazure characters.

Primality conjecture. The following conditions are equivalent:

Let $P$ be a differential poset, let $p_n$ denote the size of its rank-$n$ set, and let $U_{n-1}D_n$ be the composite of the down operator $D_n$ from rank $n$ to rank $n-1$ with th…

Let the $BK$-groups denote the groups generated by the Bender–Knuth involutions on standard tableaux, and let Krammer's groups be the groups defined in the cited presentation. Bend…

Mészáros's conjecture. After this substitution, the resulting polynomial no longer depends on the choices made during the game.

Stanley's conjecture. The $h$-vector of a matroid complex is a pure $O$-sequence.