All Questions
Tagged with algebraic-combinatorics matrices
8
questions
0
votes
1
answer
83
views
Finding an $\mathbb R$-representation for $M^*.$
Here is something I want to learn ( Problem #2.2.7 in Matroid Theory, first edition( page 88 ) , by James Oxley:
Finding an $\mathbb R$-representation for the dual matroid $M^*$ when $M$ is the vector ...
- 3,139
2
votes
1
answer
88
views
prove that $M\big[ \frac{A_1}{A_2}\big] = M[A_1].$
Here is the question I am trying to tackle:
For $i = 1,2,$ let $A_i$ be an $m_i \times n$ matrix over a field $\mathbb F.$ If every row of $A_2$ is a linear combination of rows of $A_1,$ prove that $M\...
- 3,139
5
votes
1
answer
91
views
Are there interesting combinatorial proofs which use more sophisticated grouping than sign-reversing involutions?
There are many combinatorial proofs which establish interesting identities by designing suitable "sign-reversing involutions" on a set of relevant signed objects.
For example, Benjamin and ...
- 729
2
votes
0
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82
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About the determinant of a symmetric matrix with even diagonal
Let $A$,$B$ two integer, symmetric, $n\times n$ matrix with even diagonal. We suppose also that $A$ has odd determinant (so $n$ is even).
Even discarding the hypothesis of even diagonal for $A$ it'...
- 674
9
votes
1
answer
513
views
Powers of a simple matrix and Catalan numbers
Consider $m \times m$ anti-bidiagonal matrix $M$ defined as:
$$M_{ij} = \begin{cases}
-1, & i+j=m\\
\,\,\ 1, & i+j=m+1\\
\,\,\, 0, & \text{otherwise}
\end{cases}$$
Let $S_n$ stand ...
- 26.7k
10
votes
1
answer
307
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Does in plane exist $22$ points and $22$ such circles that each circle contains at least $7$ points and each point is on at least $7$ circles.
Does in plane exist $22$ points and $22$ such circles that each circle contains at least $7$ points and each point is on at least $7$ circles.
I have solved this one but now I can't remember how I ...
- 90.7k
1
vote
2
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187
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Exercise on number of walks in a graph
The following is an exercise (Exercise #2 (a), Chapter 3, page 28) from Richard Stanley's Algebraic Combinatorics.
Let $G$ be a finite graph (allowing loops and multiple edges). Suppose that
...
- 710
9
votes
2
answers
97
views
Given a matrix $A$ such that $A^{\ell}$ is a constant matrix, must $A$ be a constant matrix?
This problem originates from an exercise in Richard Stanley's Algebraic Combinatorics. The exercise in the text (Chapter 3, Exercise 2(a)) asks
Let $G$ be a finite graph (allowing loops and ...
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