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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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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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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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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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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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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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 $B$...
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Undergrad-level combinatorics texts easier than Stanley's Enumerative Combinatorics?
I am an undergrad, math major, and I had basic combinatorics class using Combinatorics and Graph Theory by Harris et al before (undergrad level). Currently reading Stanley's Enumerative Combinatorics ...
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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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