Tagged Questions
4
votes
3answers
80 views
Dissecting an equilateral triangle into equilateral triangles of pairwise different sizes
It is know that a square can be dissected into other square such that no two of the squares have the same size.
This is the simplest dissection of that kind:
Is it also possible to dissect an ...
0
votes
1answer
25 views
Number of faces of $n$ congruent disks
If I have $n$ disks, all of the same radius, how many faces (i.e. maximally connected regions) can the induced arrangement have? For example for 3 disks, it could have 7 bounded faces, but what is ...
3
votes
0answers
44 views
Upper bounds on rate of q-ary codes
Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (MRRW) which states that the rate $R(\delta)$ ...
0
votes
1answer
74 views
A lemma regarding cones covering $\mathbb{R}^d$
Added: Pointers to some references with the same conclusion as the wolloing lemma may be helpful to understand it, and are appreciated.
In A Probabilistic Theory of Pattern Recognition By Luc ...
25
votes
1answer
836 views
Dividing a square into equal-area rectangles
How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$?
The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
4
votes
1answer
97 views
Convex hull of $n$-gon and $m$-gon
Suppose you have a convex $n$-gon, and a convex $m$-gon, in the plane. Take the
convex hull of the $n+m$ vertices. How many combinatorially distinct hulls can be obtained,
where two hulls are ...
5
votes
3answers
180 views
Number of point subsets that can be covered by a disk
Given $n$ distinct points in the (real) plane, how many distinct non-empty subsets of these points can be covered by some (closed) disk?
I conjecture that if no three points are collinear and no four ...
3
votes
1answer
126 views
Intersection of Disks
If I have a disk $d$ where each point of the disk is contained in at least $k$ other disks, then at least how many other disks does $d$ intersect?
Given, that all the disks (including $d$) have the ...
2
votes
1answer
74 views
How to calculate number of lumps of a 1D discrete point distribution?
I would like to calculate the number of lumps of a given set of points.
Defining "number of lumps" as "the number of groups with points at distance 1"
Supose we have a discrete 1D space in this ...
8
votes
1answer
252 views
“Center-of-Mass” of lattice polygons (generalization of Pick's theorem)
Call a polygon with integer coordinates (in the Euclidean plane) a 'lattice polygon'. Pick's Theorem allows you to efficiently compute the number of lattice points inside this polygon given just its ...
1
vote
0answers
94 views
Complexity of Counting the number of inducing $n$-gons
Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel.
It is clear that by extending the edges of each simple $n$-gon in ...
10
votes
2answers
389 views
Results related to The Happy Ending Problem
Im giving a small talk for a combinatorics class on the Erdos-Szekeres conjecture regarding the happy ending problem (the paper is focused on recent work regarding the conjecture). I always find that ...
