Questions tagged [co.combinatorics]
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
10,629
questions
2
votes
0
answers
53
views
How many Tverberg partition are in cloud of points?
Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect.
For example, $d=2$, $r=3$, 7 points:
Let $p_1, p_2,...
- 179
0
votes
0
answers
27
views
Closed form for the A357990 using A329369 and generalised A373183
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor, \\
\ell(0) = -1
$$
Let
$$
f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1
$$
Here $f(n)$ is A290255.
Let $A(n,k)$ be a square array such that
$$
A(n,k)...
- 4,024
-6
votes
0
answers
50
views
Power-2 cascade sequence [closed]
Definition: if a number is odd multiplied by 2 to the power k and add 2 to the power k, if a number is even divide it by two repeat this process it ends on one.
Example: let the number be 5
And k be 2
...
0
votes
0
answers
45
views
Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
Say that we draw a graph in the following way: we first draw $n$ planar embeddings of $K_{2,2}$ (that is, we first draw $n$ quadrilaterals) such there are no edges which cross. Then for each of the $...
- 1
5
votes
2
answers
280
views
Bound on the number of unit vectors with the same pairwise inner products
I want to know the bound on the number of unit vectors $v_i$ in $\mathbb{R}^n$ such that $\langle v_i, v_j\rangle=c$ for all $i\ne j$. I know this can be upper bounded by the number of equiangular ...
- 153
0
votes
1
answer
44
views
Square submatrix of a binary matrix with all columns having the same sum
Let $M$ be a $m \times n$ matrix with binary entries (i.e. a matrix all whose entries belong to the set $\{0,1\}$), with $m\geq n$. Suppose each row of $M$ contains exactly $k$ ones. Given $n$ ...
0
votes
0
answers
42
views
Tamari lattice and bicategory coherence
Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...
- 4,145
1
vote
0
answers
81
views
Evaluating the difference of weighted binomial coefficients
I encountered the following type of sum:
$$
\begin{align}
\left[
\sum_{k=1}^{t}\binom{k+i-2}{i-1}\binom{t-k+l_1-i}{l_1-i}\sum_{s=k}^{t}\binom{t-s+l_2-j+1}{l_2-j+1}\binom{s+j-3}{j-2}
\right] \tag{1} \\
...
- 11
4
votes
1
answer
358
views
4-color theorem for hypergraphs
Question. Does every hypergraph that does not admit a complete minor with $5$ elements have a coloring with $4$ colors?
Below are the definitions to make this precise.
If $H = (V, E)$ is a hypergraph ...
2
votes
0
answers
92
views
What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
- 35
0
votes
0
answers
42
views
Possible determinants of 01-matrices with at most three 1s in each row, column
As a function of $n$, what is the set of possible determinants of $n \times n$ matrices whose elements are 0s and 1s and have at most three 1s in each row and column?
I really enjoyed the problem ...
- 429
0
votes
0
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67
views
Generating function for dimensions of the space of polynomials fixed by a single permutation
Consider the space of polynomials with complex coefficients $\mathbb{C}[x_1,x_2,\dots,x_n]$ and let $\sigma$ be a permutation of $\{1,2,\cdots, n\}$ that acts on
this space via $\sigma(x_i)=x_{\sigma(...
- 141
2
votes
1
answer
113
views
Large almost disjoint family on $\mathbb{N}$ with property $\mathbf{B}$
Let
$\newcommand{\oo}{[\omega]^\omega}\oo$ denote the collection of all infinite subsets of the set of nonnegative integers $\omega$. We say that $\newcommand{\ss}{{\cal S}}\S\subseteq \oo$ ...
6
votes
0
answers
102
views
Eulerian posets and order complexes
To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its ...
- 1,949
0
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57
views
A weakening of the definition of positive roots for a root system
Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying
$$\Delta^+ = - \Delta^-\tag{$*$}\...
