Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
"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
Some confusion seems to have arisen because the question uses the notation for permutations and originally said "permutations"; this has now been changed to "sequences" (except for one instance; this seems to be an oversight). Note that all comments above were made before this change. Also, in zulon's comment, $(3,4,2,1)$ should be $(3,4,1,2)$. – joriki May 28 '11 at 7:14
@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
up vote 4 down vote accepted

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

Thanks, @joriki.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.