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I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently,...

Let us fix $\Delta$ a finite simplicial complex, and label the vertices of $\Delta$ as $\{1,2,\ldots,n\}$. For each $F\in \Delta$ let us consider the point in $\mathbb{R}^n$ given by: $$e_F := \sum_{i\...

Let $P$ be a poset, or partially ordered set. Let $\le$ denote the reflexive order on $P$, and $<$ the corresponding irreflexive order. Let the phrase "a maximal pair" in $P$ refer to an ordered ...

For a finite poset $P$ the chain polytope $\mathscr C(P)\subset\mathbb{R}^P$ consists of such $g$ that $g(p)\ge 0$ for all $p\in P$ and $$g(p_1)+\ldots+g(p_n)\le 1$$ for any chain $p_1<\ldots<...

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A convex $...