Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Algebraic combinatorics is an area of mathematics that employs methods of , notably and , in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in .

Important objects in algebraic combinatorics include Young tableaux and the ring of symmetric functions, which have connections to the representation theory of the symmetric group and of the general linear group.

For questions about algebraic graph theory or matroid theory, consider the tags or instead.

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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 before (undergrad level.) Currently reading Stanley's Enumerative Combinatorics with other math folks. We have found this book somewhat challenging~ Do you have any suggestions on…
John Dynan
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A conference uses $4$ main languages. Prove that there is a language that at least $\dfrac{3}{5}$ of the delegates know.

A conference uses $4$ main languages. Any two delegates always have a common language that they both know. Prove that there is a language that at least $\dfrac{3}{5}$ of the delegates know. Source: Romania TST 2002 My attempt: Suppose we have $n$…
nonuser
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Hartshorne Problem 1.2.14 on Segre Embedding

This is a problem in Hartshorne concerning showing that the image of $\Bbb{P}^n \times \Bbb{P}^m$ under the Segre embedding $\psi$ is actually irreducible. Now I have shown with some effort that $\psi(\Bbb{P}^n \times \Bbb{P}^m)$ is actually equal…
user38268
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Property of a polynomial with no positive real roots

The following is an exercise (Exercise #3 (a), Chapter 3, page 28) from Richard Stanley's Algebraic Combinatorics. Let $P(x)$ be a nonzero polynomial with real coefficients. Show that the following two conditions are equivalent: There exists a…
Cosima Maslani
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How to prove this $p^{j-\left\lfloor\frac{k}{p}\right\rfloor}\mid c_{j}$

Let $p$ be a prime number and $g\in \mathbb{Z}[x]$. Let $$\binom{x}{k}=\dfrac{x(x-1)(x-2)\cdots(x-k+1)}{k!} \in \mathbb{Q}[x]$$ for every $k \geq 0$. Fix an integer $k$. Write the integer-valued polynomial $\binom{g(px)}{k}$ in the…
network o
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Identity involving partitions of even and odd parts.

First, denote by $p_E(n)$ be the number of partitions of $n$ with an even number of parts, and let $p_O(n)$ be those with an odd number of parts. Moreover, let $p_{DO}(x)$ be the number of partitions of $n$ whose parts are distinct and odd. Finally,…
seungyeon
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Combinatorial interpretation of an alternating binomial sum

Let $n$ be a fixed natural number. I have reason to believe that $$\sum_{i=k}^n (-1)^{i-k} \binom{i}{k} \binom{n+1}{i+1}=1$$ for all $0\leq k \leq n.$ However I can not prove this. Any method to prove this will be appreciated but a combinatorial…
Craig
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Family of sets with $|F_i| \equiv 2\pmod 3$ and $|F_i \cap F_j| \equiv 0 \pmod 3$

Let $p$ be a prime. By considering the incidence vectors of subsets $F_1,\ldots,F_m$ of $\{1,2,\ldots,n\}$, such that $|F_i| = a \not\equiv 0 \pmod p$ and $|F_i \cap F_j| \equiv 0 \pmod p$ for all $1\leq i
DesmondMiles
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Intuition behind picking group actions and Sylow

A common strategy in group theory for proving results/solving problems is to find a clever group action. You take the group you are interested in (or perhaps a subgroup), and find some special set that your group can act on, usually by left…
jlammy
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Motivation/intuition behind using linear algebra behind these combinatorics problem

What is the motivation behind using linear algebra in these three problems ? A pair $(m,n)$ is called nice if there is a directed graph with (self edge are allowed, but multiple edge are not allowed) $n$ vertices such that for every pair of…
katana_0
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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 did it. I just remember that I used some linear…
nonuser
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why $\frac{f_n}{f_kf_{n-k}}$ is an integer for this sequence

New Zealand 2013 TST problem: Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\dfrac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$…
New Zealand
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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 for the sum of all elements of the $n$-th power of the…
user
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Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $q_{ij} = i + j -1$. Let $$H(\lambda) =…
Igor Pak
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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 multiple edges). Suppose that there is some $\ell> 0$…
Cosima Maslani
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