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Let's consider the associahedron whose vertices are triangulations of an $n$-gon. Sleator, Tarjan, and Thurston (Rotation distance, triangulations, and hyperbolic geometry, JAMS, 1988) give a simple a …

The righthand side is clearly contained in the lefthand side. We now want to show the lefthand side (i.e., $Q_I$) is contained in the righthand side. Since $Q_I$ is the convex hull of its lattice poin …

One example of an elementary application of cluster algebras is the proof that the Somos-4 and Somos-5 sequences, which are defined by a simple recursion, are integral. This is so because the entries …

I think there is good reason to think the answer is "no". In rank 2, the theta basis agrees with the greedy basis (arXiv:1508.01404). Greedy basis elements are indecomposable positive elements (see …

Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)? F …

This is a generalization of Bjørn Kjos-Hanssen's answer to your previous question. Take $K_{n+2}$ and remove two edges between two different pairs of vertices, say $(1,2)$ and $(3,4)$ The chromatic …

I think the answer is "no". Let's consider $p=2,d=3$. Suppose that we have a necklace which can be fairly divided using only 2 cuts (one less than the maximum number that may be required). Let …

Define a ground set $X$ of size $2^{n-1}$. Now choose $2^{n-1}-(n-1)$ subsets of $X$, each of size at least 2, such that the sum of their sizes is $(n-2)2^{n-1}+2$ (so the average size is slightly mo …

The answer is no. Here is a list of sets $A_i$ and $B_i$ which fails. 12, 1234 23, 1235 13, 1236 14, 1245 25, 2356 36, 1346 45, 1456 56, 2456 46, 3456 The failure can be seen by drawing the pictu …

Proposition 4 of Morelli's paper "Pick's Theorem and the Todd class of a toric variety" gives a sufficient condition: it describes a setting in which there is a positive formula for coefficient of $x^ …

Join semi-distributive lattices don't have this property because weak order on $S_n$ is join semi-distributive and doesn't have this property. (Eg, for $n=3$.) Lattices satisfying the property you a …

The following is rather more simple-minded that what you are suggesting. Let's say we have a deck of cards, with the cards labelled by the primes in order. We are going to think of constructing th …

I think the answer to your first question is "yes". Oriented matroids can be realized topologically, and I am going to use that language. (This means is that I can pretend the oriented matroid is …

It follows from the Knuth-Netto formula that the asymptotics of $I_n(k)$, for $k$ fixed, is $n^k/k!$ to first order. I claim the asymptotic behaviour of $I_n^{\sigma(y)=x}(k)$ is as $n^{k-|x-y|}/ …

Clearly there is at most one increasing path, so the only problem is to find it. Take some path, and suppose it is not increasing. So there is some length 2 subpath which is not increasing. Repla …