MathDB

Let

n \geq 2

be a natural number. For any two permutations of

(1,2,\cdots,n)

, say

\alpha = (a_1,a_2,\cdots,a_n)

and

\beta = (b_1,b_2,\cdots,b_n),

if there exists a natural number

k \leq n

such that

b_i = \begin{cases} a_{k+1-i}, & \text{ }1 \leq i \leq k; \\ a_i, & \text{} k < i \leq n, \end{cases}

we call

\alpha

a friendly permutation of

\beta

.
Prove that it is possible to enumerate all possible permutations of

(1,2,\cdots,n)

P_1,P_2,\cdots,P_m

such that for all

i = 1,2,\cdots,m

P_{i+1}

is a friendly permutation of

P_i

where

m = n!

and

P_{m+1} = P_1

Problem Statement