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# 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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### Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial $$P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k]$$ ...
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### Proof that Paley Graphs are strongly regular with parameters $(p,\frac{p-1}{2},\frac{p-5}{4},\frac{p-1}{4})$

A Paley graph is strongly regular with parameters $(p,\frac{p-1}{2},\frac{p-5}{4},\frac{p-1}{4})$. I need to prove that, and obtain the parameters too. Proving it is regular valency $\frac{p-1}{2}$ is ...
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### No triangles or rectangles in a Moore graph of diameter 2.

Can somebody explain why there cannot be any triangles or squares in a Moore graph with diameter 2? This was stated without proof in my class.
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### Proving Crapo's Lemma

Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to ...
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### Orbital dimension of the action of $S_n$ on 2-subsets

I have a question on a proof in a paper on the orbital dimension of a permutation group. Let $G \le S^\Omega$ be a permutation group. A base for $G$ is a subset $\Sigma \subseteq \Omega$ for which ...
651 views

### 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 bearing the ...
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### Maximizing Signless Stirling Numbers

Let $c(n, k)$ denote the number of permutations in $S_n$ whose cycle decomposition has $k$ cycles. For a fixed $n$, I want to find $k$ such that $c(n, k)$ is maximized. I know that the $k$ I seek is ...
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Let $n\in\mathbb{N}$. Then, for $i\in\mathbb{N}$, the $i-$th power sum if defined to be the polynomial $p_i^{(n)}:=\sum_{j=1}^n x_j^i$ in $n$ indeterminates $x_1,x_2,\ldots,x_n$. Then let $\lambda:=(\... 2answers 242 views ### Given a list of$2^n$nonzero vectors in$GF(2^n)$, do some$2^{n-1}$of them sum to 0? Let$G=(\mathbb{Z/2Z})^n$written additively,$n>1$. (you can think of it as$\mathbb{F}_{2^n}$but I didn't find that useful... yet) Let$v_i$be nonzero elements of$G$for$i \in \{1 \dots 2^n \...
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 ...
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 ...