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.
97
questions with no upvoted or accepted answers
12
votes
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answers
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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 ...
- 459
11
votes
0
answers
297
views
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. ...
- 78.5k
9
votes
0
answers
183
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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 $...
- 1,366
7
votes
0
answers
350
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Uses of Chevalley-Warning
In the recent IMC 2011, the last problem of the 1st day (no. 5, the hardest of that day) was as follows:
We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The problem asks :
Prove the ...
- 6,265
6
votes
0
answers
83
views
Proving that a group is either cyclic or not-simple
This problem is from Chapter 7 of Algebraic Combinatorics by Richard p. Stanley:
Let $X$ be a finite set, and let $G$ be a subgroup of the symmetric group, $S_X$. Suppose that the number of orbits of $...
- 549
5
votes
0
answers
601
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Suppose we have $n$ points in a plane, not all on the same line. Then they are determining at least $n$ different lines.
Suppose we have $n$ points in a plane, not
all on the same line. Then they are determining at least $n$
different lines.
Suppose points $T_1,...,T_n$ determine $m$
lines
$$\ell_i(x,y) :\;\;a_ix+b_iy+...
- 90.7k
4
votes
0
answers
144
views
Generalized Hertzsprung Problem
The Hertzsprung Problem goes as follows: In how many can we place exactly $n$ non-attacking kings on a $n \times n$ chessboard such that there is exactly $1$ king in each row and column where $n \in \...
- 862
4
votes
1
answer
248
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Why does Alon's combinatorial Nullstellensatz require working over a field.
In Alon's Nullstellensatz theorems (theorems 1.1 and 1.2 here) why is it necessary for $F$ to be a field? As far as I can tell, all the arguments in the proofs should work when $F$ is, say, an ...
- 321
4
votes
0
answers
62
views
Plethysm with Basis?
For any partition $\lambda$ we denote by $S_\lambda$ the corresponding Schur functor. Now consider $\textrm{GL}(\mathbb{C}^n)$ with its natural action on $\mathbb{C}^n$. Using character theory, one ...
- 3,615
4
votes
0
answers
118
views
Constant sum of characters
Let $q$ be a prime power and $\omega=\exp(2\pi i/q)$. For a fixed $y\in\mathbb{Z}_q^n$, the map
$$\mathbb{Z}_q^n\ni x\mapsto \omega^{x\cdot y}=\omega^{x_1y_1+\dots+x_ny_n}$$
is a character of $\...
- 450
4
votes
0
answers
78
views
Is there anywhere some explicit Bruhat decompositions are written down?
Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given.
Also, I calculated the following regarding the Bruhat decomposition of $\...
- 24.9k
3
votes
0
answers
42
views
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}...
- 3,100
3
votes
0
answers
131
views
Taylor Expansion of the Polynomial in Flajolet’s Fundamental Lemma
I am currently looking at the proof of Flajolet’s Fundamental Lemma. Before I phrase the question, I need to review the definition of $(0,k)$-path and define its weight.
Define $(0,k)$-path as the ...
- 95
3
votes
0
answers
31
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Decidability of Wilf Equivalence
I have seen a lot of papers discussing whether various permutation classes are Wilf equivalent to each other. I wonder if we could solve such problems in general with computers.
More rigorously, let $\...
- 9,080
3
votes
0
answers
69
views
Common divisor of Gaussian coefficient expressions
I have a question about common divisors of some expressions involving Gaussian coefficients, in particular in the case ${n \brack 1}_{q} = \frac{q^{n}-1}{q-1}$ where $q$ is a prime power.
It is well ...
- 13.4k