6 problems

Let $\lambda / \mu$ be a skew shape and let $\mathcal{R}_{\lambda /\mu}$ and $\mathcal{P}_{\lambda /\mu}$ be the corresponding rook matroid and lattice path matroid respectively. V…

Let $\operatorname{Pan}_{r,s,n}$ be a panhandle matroid, with $1\leq r\leq s<n$, and let $\mathcal{P}_{\operatorname{Pan}_{r,s,n}}$ be its matroid polytope. A lattice polytope is E…

Notched rectangle matroid Ehrhart-positivity conjecture. Notched rectangle matroids, equivalently cuspidal matroids, are Ehrhart positive.

Let $\mathcal{M}$ be the class of lattice path matroids. The class is efficiently pigeonhole when the relevant pigeonhole bounds can be obtained efficiently. Efficient pigeonhole c…

Let $M$ be a matroid without loops or coloops, and let $T(M;x,y)$ be its Tutte polynomial. Matroidal Merino–Welsh conjectures. The following inequalities hold: … … and … These conj…

Let $\Gamma$ be a lattice path polymatroid. Its base ring is the polynomial ring $\mathbb{C}[Y_m: m$ is a basis of $\Gamma]$, and its toric ideal is the kernel of the map … defined…