Questions tagged [algebraic-combinatorics]
For problems involving algebraic methods in combinatorics (especially group theory and representation theory) as well as combinatorial methods in abstract algebra.
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Show that a matroid $M$ is connected iff, for every pair of distinct elements of $E(M),$ there is a hyperplane avoiding both.
Here is the question I am thinking about:
Show that a matroid $M$ is connected iff, for every pair of distinct elements of $E(M),$ there is a hyperplane avoiding both.
It is in James Oxley book in ...
- 3,139
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circuit and a cocircuit can not have an odd number of common elements.
Here is the question I am trying to solve:
Show that, in a binary matroid, a circuit and a cocircuit can not have an odd number of common elements.
Here are the required definitions:
A binary matroid ...
user965463
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If $X \subseteq Y$ and $r(X) = r(Y),$ then $cl(X) = cl(Y).$
Here is the question I am trying to understand its solution:
Let $M$ be a matroid, and $r$ and $cl$ be its rank function and its closure operator. Prove the following:
$(e)$ If $X \subseteq Y$ and $r(...
- 2,087
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Proving that $r(cl(X) \cup cl(Y)) = r(cl(X \cup Y))$.
Here is the question I am trying to prove:
Let $M$ be a matroid, and $r$ and $cl$ be its rank function and its closure operator. Prove the following:
$(d)$ $r(X \cup Y) = r(X \cup cl(Y)) = r(cl(X) \...
- 2,087
1
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1
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proving that $r(X \cup cl(Y)) = r(cl(X) \cup cl(Y)).$
Here is the question I am trying to prove:
Let $M$ be a matroid, and $r$ and $cl$ be its rank function and its closure operator. Prove the following:
$(d)$ $r(X \cup Y) = r(X \cup cl(Y)) = r(cl(X) \...
- 2,087
0
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1
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92
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proving that $r(X \cup Y) = r(X \cup cl(Y)).$
Here is the question I am trying to prove:
Let $M$ be a matroid, and $r$ and $cl$ be its rank function and its closure operator. Prove the following:
$(d)$ $r(X \cup Y) = r(X \cup cl(Y)) = r(cl(X) \...
- 2,087
1
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1
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Constructing a basis for a matroid with a circuit in it.
Here is the question I am trying to solve (Matroid Theory, second edition by Martin Oxley, Chapter 1, section 2):
Prove that if $C$ is a circuit of a matroid $M$ and $e \in C,$ then $M$ has a basis $B$...
- 2,087
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The intersection of all of the flats containing $X$ equals $cl(X).$
Here is the question I am trying to solve (Matroid Theory, Second edition, Chapter 1, section 4 ):
Let $M$ be a matroid, and $r$ and $cl$ be its rank function and its closure operator. Prove the ...
- 2,087
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Sum of squares of chromatic roots of a bipartite graph
Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots ...
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Characterizing paving matroids in terms of their bases.
Here is the question I am trying to solve:
Characterize paving matroids in terms of their collections of independent sets and in terms of their collection of bases.
What exactly does it mean to ...
- 2,087
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Are there Plucker-like relations for the tensor product of two decomposable differential forms?
Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form
$$
\mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge ...
- 127
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A Question on the Pedagogical Logic Behind the Order of Two Given Exercises
In Lang's Algebra, the following two exercises are presented to the reader in the following order:
Groups Exercise 15: Let $G$ be a finite group acting on $S$, a finite set of at least $2$ elements. ...
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Subtracting four numbers in cycles
Arbitrarily give four positive numbers, assuming $A_1, B_1, C_1, D_1$;
In $A_1$ and $B_1$, subtract the smaller from the larger (0 if equal),
and the result is $A_2$; In $B_1$ and $C_1$, subtract the ...
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Why is this probability > 1?
I'm going to blow my brains out.
I have 27 cards of red, blue and green cards. There is 9 of each color. I draw 12 cards. What is the probability that I have AT LEAST 6 blues, AT LEAST 1 red and AT ...
4
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Is there a $q$-analog for the product of binomial coefficients?
The $q$-analog of the binomial coefficient $\binom{n}{k}$ may be defined as the coefficient of $x^k$ in $\prod_{i=0}^{n-1}(1+q^ix)$.
Classical arithmetic identities tend to have $q$-analogs. I am ...
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