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Let $(P,\leq)$ be a finite poset that contains a (global) minimal element $0$ and a (global) maximal element $1$. We say that a subset $U \subset P$ is upward closed if $x \in U$ and $y \geq x$ forces …

Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I wou …

If you want the first use of the term "constructible" in this context, then your reference to Mel Hochster's work is right-on. But if you want the actual notion, then things get slightly hazy. I think …

Warning: the following statement answers an older version of this question. Let $G$ be the graph you want to realize. Then, $\text{Hom}(\bullet,G) \simeq G$ where $\bullet$ is the graph containing on …

The first question is answered (modulo some details) by Forman in R Forman, Morse theory for cell complexes. Advances in Mathematics, 134 pp 90 - 145, (1998). Theorem 12.1 shows that a discrete …

Since Brendan has identified the sequence and provided values for small $n$, let me point out that the asymptotic behavior of your sequence $s(n)$ will be $$s(n) \sim \frac{1}{n!}d(n)$$ where $d(n)$ i …

It seems really hard to impose combinatorial Sperner-like conditions which would guarantee the nontriviality of the Hopf invariant. But if you allow things to get slightly more algebraic by constructi …

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial map …

Too long to fit in a comment and render all the math correctly... but why can't we just expand out $f_k(n)$ to $$ f_k(n) = \frac{n!}{n^k(n-k)!} = \prod_{j=1}^{k}\left(1-\frac{j-1}{n}\right) $$ Since …

As Gil remarks in his comment, Corollary 1 of the paper which I mentioned does not in fact imply the upper bound conjecture except when one additionally assumes isolated singularities. Still, I hope t …

Young's lattice $Y$ is a graded poset and a distributive lattice whose elements are all the partitions of $n$ for $n \in \mathbb{N}$ with the poset relation coming from inclusion of Young diagrams. He …

Here are two trivial observations while we wait for the real experts to completely solve this problem (paging Prof. Stanley...) First, note that there is a reformulation of this question that might …

I don't think a nice asymptotic formula like the one you've mentioned from Tutte's work is available for higher $g$ to the best of my understanding; it is entirely possible that someone who regularly …

Since there were no answers for a few months, I asked this question to my colleague and triangulation expert Frank Lutz. Since his response was wonderful and exhaustive, I am reproducing it here for t …

Question Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable? By efficient here I am willing to consider anything with smaller …