# 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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### Hyperplane arrangement : The Shi arrangement

I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial ...
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### For the Boolean Algebra of the subsets of $F_p^n$ find number of elements of rank $k$.

Given some "power" (that is, repeated direct products) of a finite abelian group, such as $$\mathbb{Z}_2^2=\{(0,0),(0,1),(1,0),(1,1)\}.$$ What is the number of subgroups of rank $k$. I ...
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### 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 ...
45 views

### Are reflection subgroups corresponding to closed root subsystems always parabolic?

In the third paragraph of this reference, the following is stated: let $W$ be a Coxeter group with set of roots $R$, and let $H$ be a subgroup of $W$ generated by reflections (i.e. by conjugates of ...
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### generating function of finite sum involving $\left(\!\!\binom{n}{k}\!\!\right)$

Notation: $$\left(\!\! \binom{n}{k}\!\!\right)={n+k-1 \choose k}=\frac{(n+k-1)!}{k!(n-1)!}$$ where $n!$ is the factorial, i.e. $1\cdot 2\cdots n.$ Let $n,N\in \mathbb{Z}_{>0}.$ I'm stuck at ...
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### Binomial coefficient identities from h-vector of a simplicity complex. [duplicate]

My question is to show the below equality $$\sum_{j=0}^{d}(-1)^{j}\binom{d}{j}\binom{d-j}{i}=0$$ when $d>i \geq 0$ for any integers $d,i$. This inequality is came from Stanley's note. Given an $f$-...
1 vote
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### Operations with Circulant Matrix using GAP

I am newbie using GAP software. I need to know how to use GAP software for algebraic computations with circulant matrix. Some examples would suffice. Just for clarity Circulant Matrix: In linear ...
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### Minimum value of factorial multiplication

I studied that if x + y + z = 3n (i.e if value of x+y+z is some fixed number which is multiple of 3) then minimum value of x!y!z! is when x=y=z. How can we prove this statement? Attempt: Well, I ...
1 vote
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### Change of coordinate and Linear Ode

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n$$ where $$a_n = \prod_{i=1}^{n}G((i-1)h)$$ We can conclude that the series satisfies a Linear differential ...
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### Operate an aviation network without condition: Each city has a direct flight to 3 other cities; There is always a flight visiting all cities once each

Problem. A country has $10$ cities. Operate an aviation network within $2$ following conditions: a. Each city has a direct flight to exactly $3$ other cities. b. From a beginning city, there is always ...
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### Is the poset of all subsets of a collection of sets rank-unimodal?

I'm working on Richard P. Stanley's Algebraic Combinatorics and have been stuck on this problem for a while: Let $S_1, S_2,..., S_{k}$ be finite sets with $\#S_1 = \#S_2 = ... = \#S_{k}$. Let $P$ be ...
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### Prove or disprove that there exists no more specific match-ups involving these 4 third-placed teams for the round of 16 other than

UEFA EURO 2020 has been just started, still with the format of 24 teams. The top 2 in each groups will proceed to the round of 16 along with the best of 4/6 third-placed finishers. Prove or disprove ...
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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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### In a primitive symmetric association scheme, why does $E_j$ occur in some power of $E_i$ for each $i,j$?

I am having some trouble in the proof of the Absolute Bound Condition for primitive symmetric association Schemes in the book Algebraic Combinatorics I by Bannai and Ito (Chapter 2, Section 4, Theorem ...
1 vote
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### Graph with no triangles and with two non-neighbours vertices having exactly $b$ common neighbours is regular

I have encountered the following problem: If $\Gamma$ is a graph on at least $3$ vertices and containing no triangles and with two non-neighbours vertices having exactly $b \geq 2$ common neighbours ...
77 views

### Proof of 3-perfect codes [duplicate]

I am reading a the proof of a theorem that says that $3$-perfect codes can only have length $7$ or $23$. I do not understand the following: It follows from the definition that if $n$ is the length of ...
1 vote
In a single-file queue of $n$ people with distinct heights, define a blocker to be someone who is either taller than the person standing immediately behind them, or the last person in the queue. For ...