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.
265
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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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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: ...
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3
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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(...
user38268
18
votes
1
answer
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Let $p$ be a prime number and $S\subseteq\{1,2,\cdots,p-1\}$ be a subset such that $|S|>p^{\frac{3}{4}}$...
Let $p$ be a prime number and $S\subseteq\{1,2,\cdots,p-1\}$
be a subset such that $|S|>p^{\frac{3}{4}}$. Prove that for every
positive integer $m$, there exist
$a_{1},a_{2},b_{1},b_{2},c_{1},c_{2} ...
- 90.7k
16
votes
4
answers
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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 ...
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2
answers
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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 ...
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6
answers
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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 ...
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14
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3
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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\...
- 2,853
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2
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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 ...
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12
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0
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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 ...
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11
votes
1
answer
635
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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) ...
- 1,872
11
votes
0
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Clubs whose intersections are multiples of six (Oddtown variant)
This is a question about generalizing the famous "Clubs in Oddtown" problem. The original setup is that a town has $n$ people, and $m$ clubs each consisting of a subset of the population. ...
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10
votes
1
answer
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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
10
votes
1
answer
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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}{...
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9
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2
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A question on Derangement Combinatorics
Six cards and six envelopes are numbered $1$, $2$, $3$, $4$, $5$, $6$ and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope ...
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