from itertools import permutations
name = 'sasha'
surname = 'petrov'
birth_date = '11.01.1985' # тут еще желательно разбить по точке
nickname = 'sanek'
for j in range(3):
for im in permutations(birth_date.split('.'), j+1):
for x in permutations([name, surname, nickname, ''.join(im)], 2):
print(''.join(x))
sashapetrov
sashasanek
sasha19850111
petrovsasha
petrovsanek
petrov19850111
saneksasha
sanekpetrov
sanek19850111
19850111sasha
19850111petrov
19850111sanek
sashapetrov
sashasanek
sasha19850111
petrovsasha
petrovsanek
petrov19850111
saneksasha
sanekpetrov
sanek19850111
19850111sasha
19850111petrov
19850111sanek
sashapetrov
sashasanek
sasha19850111
petrovsasha
petrovsanek
petrov19850111
...
единственно - много повторений будет
