An important exam is coming up, which will be taken by $$$n$$$ students. All students will be seated in one classroom, which has $$$n$$$ desks. Each student and desk is numbered from $$$1$$$ to $$$n$$$. In total, the students will take $$$m$$$ exams. You have been tasked with preparing a seating arrangement plan for each exam, satisfying the following conditions:

1. On any given day, all students must sit at different desks, meaning that exactly one desk should be assigned to each student.
2. Each time, the students must sit differently, meaning that there should not be two days where the seating arrangement plans are completely identical.
3. No student should sit at the same desk for consecutive exams.

The seating arrangement plan for one exam is represented by $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ $$$(1 \le a_i \le n)$$$, which means that the $$$1$$$st student should sit at desk $$$a_1$$$, the $$$2$$$nd student should sit at desk $$$a_2$$$, and so on, until the $$$n$$$th student sits at desk $$$a_n$$$.