All Questions
Tagged with enumerative-geometry co.combinatorics
12 questions
3
votes
0
answers
118
views
Divide Euclidean space by surfaces
It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is
\begin{equation}
1 + n + C^2_n + \cdots + C^k_n
\end{equation}
Is there similar ...
- 185
1
vote
0
answers
127
views
Schubert calculus and the representation ring of the general linear algebra
Schubert calculus studies the structural constants of the standard basis of the cohomology ring of the quantum Grassmannians. It is well known that it is isomorphic to the fusion ring of the category ...
1
vote
1
answer
132
views
Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon?
When I say that an $r$-sided simple (i.e., not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular ...
10
votes
1
answer
559
views
Proving Positivity for Schubert Calculus
In study of the cohomology ring of the Grassmannians, which is usually known as Schubert calculus, one usually deals with a distinguished basis known as the Schubert basis $\{\sigma_\lambda\}$. One of ...
- 643
7
votes
1
answer
293
views
Largest number of points one can pick in finite projective space without getting three on a line
Consider the projectivization $\mathbb P\mathbb F_p^n$ of $\mathbb F_p^n$. How large a set $B \subseteq \mathbb P \mathbb F_p^n$ can I pick so that no three points of $B$ lie on the same line?
- 615
2
votes
1
answer
153
views
Bounding number of $k$-nearest neighbor sets in $\mathbb{R}^d$
Suppose that $\mathcal{X} \subseteq \mathbb{R}^d$ is compact.
Let there be $n$ distinct points $X = \{ x_1,...,x_n \} \subseteq \mathcal{X}$ and $k = \lfloor n^\alpha \rfloor$ where $0 < \alpha &...
- 23
4
votes
0
answers
107
views
Random polyominoes containing $2\times2$ squares
The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...
- 13.4k
2
votes
0
answers
156
views
Enumerating the number of degree d curves tangent to a planar conic
This question is based on a special case of the Coparaso Harris formula, as described in Counting curves on rational surfaces - R. Vakil.
Let $E$ be a non-singular planar conic.
Then every degree $d$...
- 91
8
votes
0
answers
642
views
How many ways can a snake lie?
This is essentially a question about counting nonintersecting short paths in a
cubic lattice, but with a twist. (One constraint that I did not make clear below
is that when to turn is already chosen:...
- 2,158
6
votes
1
answer
599
views
Embedding $G(2,n)$ into $G(k,n)$
Let
$$M=\begin{pmatrix}
u_1 & u_2 & \ldots & u_n \\
v_1 & v_2 & \ldots & v_n \\
\end{pmatrix}$$
be a $2 \times n$ matrix. Define $\nu(M)$ to be the $k \times n$ matrix
$$\nu(M)...
- 156k
12
votes
3
answers
1k
views
Counting restricted polyominoes
I would like to count the polyominoes of $n$ squares that satisfy several restrictions:
Each is convex: every horizontal, or vertical line, meets the shape in either a single segment,
or not at all.
...
6
votes
2
answers
451
views
Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?
Background
I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More ...
- 95