In my opinion, for n=4 the correct answer is 20 if I understood the condition of the problem correctly, i.e for n=3 there are exists 2 ways:
123231 and 132321. Please, explain me my mistake if I'm wrong.

12324341
12342341
12342431
12343241
12343421
12423431
12432341
12432431
12434231
12434321
13234241
13242341
13242431
13243241
13243421
13423241
13423421
13424231
13424321
13432421
14232341
14232431
14234231
14234321
14243231
14323241
14323421
14324231
14324321
14342321

Oh! Now I understood my mistake. Thank you!