# Questions tagged [permutation-cycles]

For elementary questions concerning permutation cycles and permutation groups. This includes all representations of permutations (two-line arrays, cycles, bipartite graphs); transpositions and the sign/parity of a permutation; the Symmetric group and the Alternating group. To be used with the (permutations) tag to make the distinction between abstract algebra permutation questions and combinatoric permutation questions.

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### Coloring $5 \times5$ transparent chessboard in space.

We have $5 \times 5$ chessboard floating in space , so rotation and reflection are allowed. We want to color it using $m$ distinct colors such that when we color a square , the both sides of that ...
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### Placing red and black colors on $2 \times 4$ chessboards

Suppose that two chessboards are also considered equivalent (aside from rotational symmetry) if one can be obtained from the other by complementing red and black colors. How many different $2 × 4$ ...
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### 4 different colour balls, each of number four have to be arranged in a circular manner so that adjacent 3 balls are of different colour.

I have 4 red balls, 4 green balls, 4 Blue balls and 4 Yellow balls with me. I have to arrange them in a circular manner. The condition is that if we take any 3 adjacent balls, they should have to be ...
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### Subgroup of $A_n$ generated by a 3-cycle and an n-cycle

As pointed out here, an $n$-cycle $a=(12\ldots n)$ and a 3-cycle $b=(147)$ won't generate $A_n$ if n is an odd multiple of 3, at least for $n=9$. How do we calculate the structure and order of this ...
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### Faces of soccer/football as cycles of vertices?

In this question, I asked for help finding the adjacency matrix for a truncated icosahedron. (There are two good answers there). Now I have a related question. Is there a resource for listing the ...
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### Prove that $\Gamma_{abc}=\frac{1}{2}\left(\partial_bg_{ac} + \partial_cg_{ab}-\partial_ag_{bc}\right)$

I am tasked with the following problem Use the equation $$\nabla_ag_{bc}=\partial_ag_{bc}-\Gamma_{cba}-\Gamma_{bca}=0\tag{1}$$ where $$\Gamma_{abc}=g_{ad}\Gamma^d_{bc}\tag{A}$$ and the (no torsion) ...
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### Pseudorandom permutation of 60000 elements with a long period

I have a programming assignment that asks me to do mini-batch training. In particular, we are working with the MNIST dataset, which contains 60000 training samples. I would like to figure out the most ...
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