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Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data: a noncrossing matching on $2n$ vertice …

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 …

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 …

This is "the same" as the generating function proof, but it doesn't use generating functions explicitly. Take the largest common difference in any of the sequences, say n, and pick $\zeta$ a primitiv …

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 …

A better solution than my previous one is max_{1\leq i \leq n} iN - {i \choose 2}N_2 (That is to say, we can simply consider only i of the sets instead of all n of them, and then apply my previous …

It seems unlikely (once you assume d.c.c.). Define the height of an element $x$ in $P$ to be the length of the shortest unrefinable chain from $x$ to $0$. Let $P_n$ denote the elements of $P$ whos …

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 …

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 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 …

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 …

This is an answer to Alexander's combinatorial reformulation of the question in comments to Bruce's answer. dim $V_\lambda$/$n$! is the chance that you will get a standard Young tableau if you assi …

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 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 …

O(1) is impossible even if you drop the condition of no monochromatic rectangles, and even if you know that the two cells are always chosen within a given row. Suppose the length of the rows, $m$, …