How many permutations $(a_i)_{i=1}^{30}$ of $\{1,\dots,30\}$ satisfy $m$ divides $a_{n+m}-a_n$ when $m \in \{2,3,5\}$ and $1 \le n

Question

Define a permutation $(a_1,a_2,\dots,a_{30})$ of $\{1,2,\ldots,30\}$ as good if for all $m \in \{2,3,5\}$, we have that $m$ divides $a_{n+m}-a_n$ for all integers $n$ satisfying $1 \leq n < n+m \leq 30$. How many good permutations are there?

I don't understand how to formulate the binomial expressions and casework. Could someone provide me with a solution? Thank you.

The question is getting votes to close as unclear; I've reworded it with the intention to make it clearer. But please [edit] if I've blundered. — Rebecca J. Stones, Jul 25 '15 at 05:00

score 1 · Accepted Answer · answered Jul 25 '15 at 05:05

1

Hint: the condition on $m=2$ says either all the even numbers are in even positions or all the even numbers are in odd positions. What does $a_1 \bmod 3$ imply? How about $a_1 \bmod 5$?

answered Jul 25 '15 at 05:05

Ross Millikan

374,822

I understand that the there is a bijection between the indices and the corresponding numbers moduli, however do you mind providing a solution, as I still can't get much further. Thank you. – Saketh Malyala Jul 25 '15 at 19:16

How many permutations $(a_i)_{i=1}^{30}$ of $\{1,\dots,30\}$ satisfy $m$ divides $a_{n+m}-a_n$ when $m \in \{2,3,5\}$ and $1 \le n

1 Answers1