18 problems

Möbius support conjecture. For any base polymatroid $\mathscr{P}$, the Möbius support $\mu\text{-supp}(\mathscr{P})$ is a generalized polymatroid. The theorem preceding this conjec…

Let $P$ be the base polytope of a polymatroid. A triangulation of $P$ is unimodular if all its simplices are unimodular with respect to the relevant lattice, and it is flag if ever…

Castillo–Cid-Ruiz–Mohammadi–Montaño conjecture. The Möbius support of $\mathscr{P}$ is a generalized polymatroid, that is, a homogenization of it yields a polymatroid. The paper se…

Let $R=\mathbb{k}[x_1,\ldots,x_p]$ be a polynomial ring over a field $\mathbb{k}$, and let $\mathscr{P}$ be a polymatroid on $[p]$ with cage $\mathbf{m}\in\mathbb{N}^p$. Let…

Symmetric exchange conjecture. The toric ideal

Let $E$ be a finite set, let $\mathrm{P}_1$ and $\mathrm{P}_2$ be polymatroids on $E$, and let $B(\mathrm{P}_1)$ and $B(\mathrm{P}_2)$ denote their base polytopes. Suppose that … F…

The hierarchical-order conjecture. If

Let $\mathcal A$ be an access structure. It is ideal if it can be realized by an ideal entropic polymatroid, equivalently by an ideal distribution scheme. Let $\mathcal A^\bot$ den…

Let $P$ be a participant set, let $\mathcal A\subset 2^P$ be an access structure, and let $\mathcal A^\bot$ denote its dual access structure. Let $\sigma(\mathcal A)$ denote the en…

Let $M$ be a connected $k$-polymatroid and let $N$ be a connected minor of $M$. A connected minor-preserving deletion or contraction is a deletion or contraction that remains conne…

Sticky polymatroid conjecture. Every sticky polymatroid is modular.

Let $E$ be a finite set, let $\mu$ be a unitary connectivity function on $E$, and let $\mu\downarrow X$ denote the connectivity-function minor obtained by the operation defined in…

Let $E$ be a finite set, and let $u$ be the connectivity function on ${a,b,c,d}$ defined by … … and with all other values determined by symmetry. A connectivity function is unitar…

Let $h$ be a polymatroid function on a lattice. The entropicity conjecture. If $h$ satisfies all Ingleton inequalities and all strong union inequalities, then $h$ is entropic. The…

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…

Let $\operatorname{PolyMat}$ be the vector space of polymatroids, let $\operatorname{QSym}$ be the quasi-symmetric-function space, and let…

Let ${\mathcal G}$ be the invariant of polymatroids defined in the paper, and let $g_{\bf X}(t)$ be Speyer's invariant of a matroid ${\bf X}$. Specialization conjecture. The invari…

Let ${\bf X}=(X,\operatorname{rk})$ be a polymatroid, and let ${\mathcal H}[{\bf X}]$ and ${\mathcal P}[{\bf X}]$ denote the invariants appearing in Corollary … should remain true…