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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Exercise 11.1 in Algebraic Combinatorics by Stanley
Let $\mathcal{C}_{D} = \mathcal{C}$ denote the cycle space, or the space of all circulations, on some digraph $D$. Let $C_n$ denote the $n$-dimensional hypercube, and $\mathfrak{o}$ denote some ...
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Exercise 2.2 In Stanley's Algebraic Combinatorics
This is Exercise $2.2$ In Stanley's Algebraic Combinatorics. I don't have much work to show because despite being stuck on this problem for a long time, I haven't got a clue how to start.
$\mathcal{C}...
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Exercise 9.6 in Algebraic Combinatorics by Stanley
Exercise 6 in chapter 9 of Algebraic Combinatorics by Stanley: Let $G$ be a finite graph on $p$ vertices with Laplacian matrix $L(G)$. Let $G'$ be obtained from $G$ by adding a new vertex $v$ and ...
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Regarding the number of variables in Symmetric Functions
I'm studying Symmetric Functions and I came across a doubt that could be considered stupid but I need clarifications.
In the course I'm following we introduced symmetric functions as formal series of ...
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Exercise 8.6 of Algebraic Combinatorics by Stanley
Problem 6 in Chapter 8 of Algebraic Combinatorics by Stanley: Show that the total number of standard Young tableaux (SYT) with $n$ entries and at most two rows is ${n \choose \lfloor n/2 \rfloor}$. ...
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Exercise 7.2 in Algebraic Combinatorics by Stanley
This is exercise 2 in chapter 7 of Algebraic Combinatorics by Stanley.
For part (a), I first found the entire automorphism group. By labeling the root 1, and then numbering off the remaining vertices ...
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Exercise 3.1 of Algebraic Combinatorics by Richard Stanley
Exercise 3.1: Let $G$ be a (finite) graph with vertices $v_1, \ldots, v_p$. Assume that some power of the probability matrix $M(G)$ defined by $(3.1)$ has positive entries. (It's not hard to see that ...
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find the general solution the recurrence equation $b_n = 3b_{n-1} - b_{n-3}$
here are the steps I have done to try and find the general solution of this relation:
$$ b_n = 3b_{n-1} - b_{n-3}\\ = b^n = 3b^{n-1} - b^{n-3}$$
then divide by $b^{n-3}$ to get $$b^3 = 3b^2 - 1$$
then ...
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find the number of ways to distribute 30 students into 6 classes where there is max 6 students per classroom
here is the full question:
Use inclusion/exclusion to find the number of ways of distributing 30
students into six classrooms assuming that each classroom has a maximum capacity
of six students.
Let $...
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Identity of Schur polynomials
Let $p_n$ be the power sum symmetric polynomial,
$$p_n=x_1^n+x_2^n+\dots x_n^n$$
in $n$ variables, and let $s_\lambda$ be the Schur polynomials. I am new to Schur polynomials so I'm not sure what ...
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create a recurrence relation for the number of ways of creating an n-length sequence with a, b, and c where "cab" is only at the beginning
This is similar to a problem called forbidden sequence where you must find a recurrence relation for the number of ways of creating an n-length sequence using 0, 1, and 2 without the occurrence of the ...
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Isometric automorphisms of the ring of symmetric functions
I was trying to understand how special the $\omega$ involution on the ring of symmetric functions $\Lambda$ or $\Lambda^n$ (restriction to $n$ variables, just in case if by some magic, the situation ...
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Composition of homomorphisms of association schemes
In Zieschang's "Theory of Association Schemes", in section 5.2, he remarks that the composition of homomorphisms is not always a homomorphism. I've been struggling to find an example of that ...
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Restriction of induced representation over a Young subgroup and Littlewood-Richardson coefficients
I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\...
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Affiness, $U_{2,4}$ and $M(K_4).$
I do not know why $M(K_4)$ is not affine over $GF(2)$ or $GF(3)$ but it is affine over all fields with more than 3 elements. I proved that $U_{2,4}$ is $\mathbb F$-representable iff $|\mathbb F| \geq ...
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Lift and frame matroids.
I want to read more about lift matroid and frame matroid and their flats and relations to signed graphs, do you know any basic resources for this?
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Characteristic polynomial and bounded regions.
I know that the number of bounded regions of a homogeneous hyperplane arrangement $\mathcal{A}$(a collection of n hyperplanes) in $\mathbb R^d$is ${ n - 1 \choose d}$ but how can this be expressed in ...
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Understanding contraction in hyperplane arrangements.
Here are two figures that shows a hyperplane arrangement's contraction (this is from McNulty book, "Matroids, a geometric introduction"):
I am not sure why a became a line in the right ...
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Hyperplane Areangements and contraction.
I am trying to understand an idea presented in McNulty book, matriods a geometric introduction about the new hyperplane arrangement $\mathcal{A}^{''} = \{ H \cap H_x | H \in \mathcal{A}\}$ where $\...
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No minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I am trying to find all the matroids having no minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I know that the matroid $U_{0,n}$ is a matroid of rank zero with n edges and so it is just a vertex with n ...
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Contraction, loops and flats.
This idea is being used a lot, but I cannot justify why it is correct:
If M is a matroid and $T$ a subset of $E(M).$ Then $M/T$ has no loops iff $T$ is a flat of $M.$
I know how to proof that in a ...
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The basis of a regular matroid.
I know that a regular matroid is one that can be represented by a totally unimodular matrix.
I also know that a rank r totally unimodular matrix is a matrix over $\mathbb R$ for which every submatrix ...
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affine geometries that are self-dual matroids.
I want to know which of the affine geometries $AG(n,q)$ are self-dual matroids?
I have proved before that uniform matroids $U_{n,m}$ that are self duals are those who satisfies that $m = 2n.$ I am ...
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Understanding how to find the dual of a matroid.
I am trying to understand the following picture of a matroid and its dual but I found myself not understanding exactly what we are doing to find the dual:
]1
Roughly speaking, according to some ...
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what will happen if we contract an element in a uniform matroid? [closed]
Are the parallel elements in a matroid just behaving like loops? If so, why?
For example, in $U_{2,3}$ if we contract an element what will happen? In $U_{2,2}$ if we contract an element what Will ...
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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- $\beta(M) > 0$ ...
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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 ...
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Show that $C^*(e,B)$ is the unique cocircuit that is disjoint from $B - e.$
I want to prove the following question:
Let $B$ be a basis of a matroid $M.$ If $e \in B,$ denote $C_{M^*}(e, E(M) - B)$ by $C^*(e,B)$ and call it the fundamental cocircuit of $e$ with respect to $B.$\...
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Showing the isomorphism between (the geometric lattice)$\pi_n$ and the lattice $\mathcal{L}(M(K_n)).$
Here is the question I am trying to solve:
Show that the lattice of flats of $M(K_n)$ is isomorphic to the partition lattice $\pi_n$. Definition: $\pi_n$ is the set of partitions of $[n]$, partially ...
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if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.
here is the question I am trying to solve:
In a matroid $M,$ if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.
I know how to prove that a set ...
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