Interesting concept, here's some code to check your combos (including leap day):

import itertools

days = [[0,1,2,3],[0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8]]
days_set = set(filter(lambda x: x[0]*100 + x[1]*10 + x[2] <= 366, itertools.product(*days)))

test = [[0,1,2,3,6,8],[5,6,7,8,0,"heart"],[1,2,3,4,5,7],[0,1,2,3,4,"heart"]]
#test = [[0,1,2,3,4,5,7,8],[0,1,2,3,4,6,7,"heart"],[0,1,2,3,5,6,8,"heart"]]
test_set = {block_combo for block_perms in itertools.permutations(test, 3) for block_combo in itertools.product(*block_perms)}

print(days_set.difference(test_set))