19
votes
Accepted
Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space
This configuration has automorphisms by the symmetric group $S_5$,
and can be identified with the planes $a_i = a_j$ ($0 \leq i < j \leq 4$)
in the projective 3-space $a_0+a_1+a_2+a_3+a_4 = 0$, by ...
- 79.9k
14
votes
Accepted
What are Sylvester-Gallai configurations in the complex projective plane?
Yes, there are other Sylvester-Gallai configurations in $\mathbb{P}^2(\mathbb{C})$. Apart from the Hesse configuration (that contains $9$ points) the minimum number of points for a non-collinear ...
- 66.3k
13
votes
Seeking a more symmetric realization of a configuration of 10 planes, 25 lines and 15 points in projective space
With the aid of the answer I now managed to find a better realization. At the expense of some of the symmetries it can be nicely drawn in 3d space - it is just the barycentric subdivision of a ...
12
votes
Accepted
Chromatic number of a graph defined by $n$ lines on the plane
No, here is an example that needs four colours (if I have understood the question correctly):
(there are other intersection points not shown, of course, but these are irrelevant)
- 7,013
11
votes
Accepted
Hyperplane arrangements whose regions all have the same shape
This is a known open problem (for isometric regions), which, as far as I know, is still not settled.
The dimension 3 case was proved affirmatively in https://arxiv.org/abs/1501.05991, where also some ...
- 3,271
10
votes
What are Sylvester-Gallai configurations in the complex projective plane?
I asked around about this question a while ago and the best answer I got was from Konrad Swanepoel. There are the well-known "Fermat" examples
$$(x^n - y^n)(y^n - z^n)(z^n - x^n) = 0, \qquad n \ge 3....
- 82.6k
9
votes
Number of regions formed by $n$ points in general position
This is more of a long comment than an answer. It should be possible
to compute the number of regions and number of bounded regions using
Whitney's theorem for the characteristic polynomial $\chi(t)$ (...
- 50.8k
9
votes
Accepted
Complexity of counting regions in hyperplane arrangements
The problem is $\#\mathsf{P}$-complete. As you already noted, the problem is $\#\mathsf{P}$-hard even when we restrict to graphical arrangements, so it remains to show that the problem is in $\#\...
- 82.6k
8
votes
Bijection directly from (n,n+1)-core partitions to parking functions?
Increasing parking functions are in (more or less canonical) bijection with Dyck paths (see, e.g., here), so your question can be rephrased as
Is there a direct bijection between (n,n+1)-cores and ...
- 3,271
8
votes
Accepted
Simplicial set represented by an (unordered) set
You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^\bullet(X)$ is infinite dimensional if $X$ ...
- 55.9k
8
votes
Accepted
Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
Here is an example with $n=4$. (ADDED BELOW: An example for any power of $2$)
I identify $\mathbb R^4$ with the quaternions, and describe $10$ subspaces such that any two of the resulting orthogonal ...
- 55.9k
7
votes
Accepted
Seeking combinatorial interpretation of a quantity that comes up from central hyperplane arrangements
Let $\Delta$ be a matroid complex, i.e., an abstract simplicial complex whose faces are the independent sets of a matroid $M$. Let $K[\Delta]$ denote the face ring (aka "Stanley-Reisner ring"...
- 50.8k
6
votes
Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
If there is a Hadamard matrix of size $n/2$, then there is a set of orthogonal matrices as desired. Recall that Hadamard's conjecture predicts that there is a Hadamard matrix of size $m$ whenever $m \...
- 156k
6
votes
Accepted
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
For $d=2$ the maximal $n$ is $3=d+1$. Indeed, between 4 points $x_1,x_2,x_3,x_4$ either one of them, $x_k$, belongs to the convex hull of three others, then $\langle \theta,x_k\rangle$ can not be the ...
- 109k
5
votes
Accepted
Singularities at worst like a hyperplane arrangement
This paper uses the term "arrangement of smooth, complex algebraic hypersurfaces", or simply, "arrangement of smooth hypersurfaces". To quote: "Our goal here is to further generalize these results to ...
- 2,163
4
votes
Accepted
The characteristic varieties of the complement of the braid arrangement
Let $T$ be an irreducible component of $V^1_d(X_k)$ with $d\ge 2$. If $T$ contains $\mathbb{1}$, then $T=\{\mathbb{1}\}$. It is probably the case that all components of $V^1_d(X_k)$ pass through the ...
- 2,163
4
votes
Bijection directly from (n,n+1)-core partitions to parking functions?
After Christian Stump's restatement of your question, in the language of Dyck paths, let's one more reference here which extends the discussion to multi-core partitions, posets and lattice paths (with ...
- 43.2k
4
votes
a Littlewood–Offord-type problem concerning the "cubical lattice"
Here is a simple proof in the case when $K$ has characteristic $2$.
Let $m = \frac{n}{2}$.
For $0\le i < m$, and $x\in K^n$, let $p_i(x)=(x_{2i},x_{2i+1})$.
I claim that for any fixed $0\le i < ...
- 3,446
4
votes
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
This is not a complete answer, but too large to fit reasonably as a comment. As shown by Fedor Petrov, it is not true that $n \leq d+1$, since for $d = 3$ we can construct $5$ such points and ...
- 716
3
votes
Accepted
Extensions of combinatorially equivalent hyperplane arrangements
I'm pretty sure you can do this with line arrangements:
The two black line arrangements are equivalent, but in the picture on the right if we add two parallel lines that intersect only at triple ...
- 24.2k
3
votes
Number of regions formed by $n$ points in general position
Perhaps it is worth quoting this theorem, even though
it does not distinguish bounded from unbounded cells,
and is phrased in terms of the number of hyperplanes
rather than the number of points ...
- 151k
3
votes
Affine Hyperplane Arrangements in $\mathbb R^d$
A naive way to find representatives is to solve $2^m$ systems of linear inequalities. More precisely, you probably want interior points, so your
inequalities will be of the form $\langle u_i,x\rangle \...
- 14k
3
votes
Counting Regions in Hyperplane Arranglements
Suppose we have $n$ sets of $r$ parallel hyperplanes in $\mathbb{R}^d$
in generic position. There are $r^k\binom nk$ ways to choose $k$ of
them that intersect in a flat $x$. The interval from $\hat{0}$...
- 50.8k
3
votes
Accepted
The Salvetti complex of a non-realizable oriented matroid
I think, the reference you are looking for is the paper by Björner and Ziegler "Combinatorial Stratification of Complex Arrangements" in JAMS.
They give a complete combinatorial proof in Sec....
- 430
2
votes
Bijection directly from (n,n+1)-core partitions to parking functions?
I believe I have as much of an answer as I'm going to get. An $(n,n+1)$-core turns into an $n$-abacus diagram (this map is at the heart of Anderson's paper Partitions which are simultaneously t1- and ...
- 525
2
votes
Counting Regions in Hyperplane Arranglements
Use Radon's theorem
to show that homogeneous hyperplanes $w$ can shatter (i.e., assign all possible sign sequences via $x\mapsto\text{sign}(<w,x>)$ at most $d$ points.
This is an upper bound on ...
- 6,541
2
votes
Accepted
Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history
Combinatorial types of hyperplane arrangements are called dissection types in the dissertation of L. Finschi: https://finschi.com/math/publ/2001-08-31_Finschi_A-Graph-Theoretical-Approach-for-...
2
votes
Rigid line arrangements
Perhaps you are familiar with this survey, which seems relevant:
Felsner, Stefan, and Jacob E. Goodman. "Pseudoline Arrangements." Handbook of Discrete and Computational Geometry, JE Goodman, ed., ...
- 151k
2
votes
Number of regions formed by $n$ points in general position
This non-answer completes Joseph O'Rourke's nice non-answer, for the case of $n$ hyperplanes in $\mathbb{R}^d$ in general position. But it also suggests that the OP situation may also well have ...
- 5,103
2
votes
Action of Weyl group on regions of Shi arrangement
I have what's maybe not an answer but is I hope a helpful comment for this question. I have a way to show that the "orbits" you describe here are in bijection with a set of cosets of a ...
- 525
Only top scored, non community-wiki answers of a minimum length are eligible