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Consider the set, $S$, of $n$-tuples defined inductively as follows:

  • $(1, 2, \ldots, n) \in S$
  • if $(x_1, x_2, \ldots, x_i, x_{i+1}, \ldots, x_n) \in S$, then $(x_{i+1}, \ldots, x_{n}, x_1, x_2, \ldots, x_{i}) \in S$

What is the name of these types of $n$-tuples?

Note, the $n$-tuple $(2, 4, 1, 3)$ demonstrates that $S$ is a strict subset of all permutations.

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I'm not sure, but it seems that $S$ contains only one element, namely $(1,2,\dots,n)$. Indeed $(1,2,3,4) = (2,3,4,1) = (3,4,2,1) = (4,1,2,3)$. Could you clarify that? – Najib Idrissi May 28 '11 at 6:20
"Note the sequence ... subset of all permutations." consider the singleton, or the pair. Then $S$ is exactly all the permutations. – Asaf Karagila May 28 '11 at 6:23
@zulon -- I am using parenthesis to express a sequence, so $(1, 2) \neq (2, 1)$. – dsg May 28 '11 at 6:26
@Asaf Karagila -- You are correct for $n = 1, 2, 3$ it is true that $S$ is the set of all permutations. But when $n \geq 4$, I believe it is a subset. – dsg May 28 '11 at 6:27
@dsg: You're using notation that's usually used for permutations and $n$-tuples; I think it would be clearer to refer to these things as $n$-tuples. Also, "inductively" is slightly misleading, since all elements of $S$ can be generated by a single application of the second rule with suitable $i$; subsequent applications just generate the same tuples again. I would call the tuples in $S$ the cyclic permutations of the tuple $(1,2,\dotsc,n)$. – joriki May 28 '11 at 7:17

1 Answer 1

up vote 4 down vote accepted

These are cyclic permutations of the tuple $(1, 2, \ldots, n)$.

Thanks, @joriki.

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