All Questions
Tagged with algebraic-combinatorics combinatorial-geometry
10
questions
2
votes
1
answer
87
views
Partition lattice properties and an invariant.
I am trying to guess the value of the beta invariant of the partition lattice $\pi_4$ if I know the following information:
For any matroid $M,$ I know that
1- $\beta(M) \geq 0.$
2- $\beta(M) > 0$ ...
- 95
1
vote
1
answer
82
views
what will happen to the uniform matroid $U_{2,m}$ if we remove an element from it?
I am trying to figure out what will happen to the uniform matroid $U_{2,m}$ if we remove an element e from it, where e is neither a coloop nor a loop. I am guessing that it will become disconnected ...
- 95
0
votes
1
answer
71
views
Why always the Crapo beta invariant value greater than or equal zero?
Here are the definitions of the Crapo beta invariant I know:
My definition of the Crapo's beta invariant of a matroid from the book "Combinatorial Geometries" from page 123 and 124 is as ...
- 3,139
2
votes
1
answer
61
views
an $\mathbb F$- representation of the matroid $M_1 \oplus M_2.$
Suppose that $A_1$ and $A_1$ are $\mathbb F$- representations of the matroids $M_1$ and $M_2$ respectively, show that
$$\begin{matrix}
A_1& 0\\
0 & A_2
\end{matrix}$$
is an $\mathbb F$- ...
- 3,139
5
votes
0
answers
601
views
Suppose we have $n$ points in a plane, not all on the same line. Then they are determining at least $n$ different lines.
Suppose we have $n$ points in a plane, not
all on the same line. Then they are determining at least $n$
different lines.
Suppose points $T_1,...,T_n$ determine $m$
lines
$$\ell_i(x,y) :\;\;a_ix+b_iy+...
- 90.7k
1
vote
0
answers
38
views
On subsets of $\mathbb N^2$ with elements not comparable w.r.t. componentwise order
Let $\mathbb N$ denote the set of nonnegative integers . For $(a,b);(c,d)\in \mathbb N^2$, define $(a,b)\le (c,d)$ iff $a\le c$ and $b\le d$. Let us call call a subset $S\subseteq \mathbb N^2$ to ...
- 4,454
3
votes
1
answer
83
views
On finding a finite set of generators for a certain semigroup
Let $A$ be a finite subset of $\mathbb Z^2$. Let $\mathbb ZA$ be the subgroup of $\mathbb Z^2$ generated by $A$. Let $\mathbb R_{+}, \mathbb Q_{+}$ denote the set of non-negative real and rational ...
- 1,443
7
votes
3
answers
526
views
How many $n$-pointed stars are there?
Say we have $n$ distinct points spaced evenly in a circle. Define a star as a connected graph with these points as vertices and with $n$ edges, no two having the same endpoints. We think of two stars ...
- 113
2
votes
1
answer
77
views
What breaks down in the theory of affine hyperplane arrangments?
It appears to me that there is a substantial amount of combinatorial algebra and geometry supporting the theory of central hyperplane arrangements (See Topics in Hyperplane Arrangements, Aguiar and ...
- 613
4
votes
1
answer
160
views
Prescriptive version of counting hyperplane arrangements
In Hyperplane arrangement theory, Zaslavsky's Theorem necessarily bounds
the number of bounded and unbounded regions in the complement of a real hyperplane arrangement. While this counting theorem is ...
- 613