# 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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### 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 ...
2answers
40 views

### 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 ...
3answers
107 views

1answer
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### Merging polynomials together

Given the (monic) polynomials $P$ and $Q$, we can split them over an appropriate field into linear factors to write $P(x)=(x-a_1)\dotsb(x-a_m)$ and $Q(x)=(x-b_1)\dotsb(x-b_n)$, and then form the ...
1answer
68 views

0answers
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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 ...
1answer
57 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 ...
1answer
669 views

### Single file queue with people as blockers

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 ...
0answers
32 views

### RSK and Matrices

It is well known that the RSK algorithm assigns to every square matrix with nonnegative integer entries a pair of semistandard Young Tableaux of same shape. The matrices are here used as just a square ...
0answers
81 views

### Number for ways to arrange 10 distinct object into 4 distinct boxes

What is the number of ways to arrange 10 distinct objects into 4 distinct boxes, where each box hold no more than 4 objects This question was asked earlier today, but has been deleted, I don't know ...
0answers
34 views

### On subsets of $\mathbb N^2$ with elements not comparable w.r.t. componentwise order

Let $\mathbb N$ denote the set of nonnegative integers . For $(a,b);(c,d)\in \mathbb N^2$, define $(a,b)\le (c,d)$ iff $a\le c$ and $b\le d$. Let us call call a subset $S\subseteq \mathbb N^2$ to ...
0answers
26 views

### Postnikov's Lemma 16.3 in TOTAL POSITIVITY, GRASSMANNIANS, AND NETWORKS

Paper linked here: https://math.mit.edu/~apost/papers/tpgrass.pdf I'm having trouble parsing the proof of Lemma 16.3, perhaps because I only have first-principles-level familiarity with matroids. I ...
1answer
326 views

0answers
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### Counting covered pairs of integer partitions

I am trying to solve a problem in algebric combinatorics, where i want to prove that for integers $m,n$ and pairs ($\lambda$ , $\mu$) of integer partitions with a maximum of $m$ non zero parts which ...
2answers
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### Combinatorics in a $n \times n$ grid

Natural number $n>2018$ is given. Numbers $1,2,\ldots,n^2$ are written (in an arbitrary order) into the fields of the $n\times n$ grid. Prove that it is possible to choose $n$ fields so that there'...
0answers
50 views

### Combinatorial proof of the identity relating Hurwitz numbers

Let define $H^{m}((n);\mu)$ count the number of tuples $(\alpha,\tau_1,\ldots,\tau_r ,\beta)$ in symmetric group $S_n$ where $\alpha$ is fixed cycle of type $(n)$ and $\beta$ cycle of type $\mu$ and ...
4answers
177 views

### Algebraic proof of a combinatoric question (Combinatoric proof is given)

I had a IMO training about double counting. Then, there is a problem which I hope there is a combinatoric proof. Here comes the problem: For every positive integer $n$, let $f\left(n\right)$ be ...
0answers
49 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 ...