9 problems

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 $S_M=k[y_B:B$ 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…

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…

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

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…