If we plot the scores of the cows in a pair-off that FJ wins on the number line, then:
- If, at a point on that number line, FJ has more cows at or before that point than the amount of cows that FP has before that point, then FP has more cows after that point than FJ, so, assuming that FJ has N cows after that point, FJ’s (N+1)th-highest-scoring-cow loses to FP’s (N+1)th-highest-scoring-cow, so the amount of cows that FJ has at or before that point must be less than or equal to the amount of cows that FP has at or before that point.
- K cows in that pair-off must be FJ’s, and K cows in that pair-off must be FP’s , so the amount of cows in that pair-off that are FJ’s must be equal to the amount of cows in that pair-off that are FP’s .
In other words, FJ’s cows must balance with FP’s cows in the same way that ) must balance with ( .
If we sort the cows first by score and then by whether FJ owns that cow, then the list of cows is equivalent to a string of parentheses where:
- a cow that belongs to FJ corresponds to a )
- a cow that belongs to FP corresponds to a (
- the parentheses appear in the same relative order as the cows that they correspond to
.
The question becomes, “given a string of parentheses and some K, how many ways can you make a string of length 2K, of balanced parentheses, and such that that string is a subsequence of that given string?”.
I wrote a Python script that attempts to solve this problem first by generating a list of all the ways to balance K open parentheses, and then to solve the problem. I know for sure that that routine to generate all the ways to balance K open parentheses – balancedArrangements – works, and I think that the routine that should find the amount of times a given string appears as a subsequence in another string – nwaysToGetArrangement – works, but the program is counting 517 winning teams for the given test case, whereas USACO calculates 382 winning teams for that test case…
nwaysToGetArrangement is not double-counting, and seems to be properly finding subsequences (print(f"\t{match}") was replaced with print(f"{''.join([')' if cows[icow].ofFJ else '(' for icow in match])}") and all of the sequences of parentheses do match up with the sequence that they are trying to find), and I cannot figure out why the program is wrong, unless my reduction of the problem is wrong, but I do not know why that reduction would be wrong…
Please help 
My Python code is below
from functools import cache #I know that USACO doesn't allow functools.cache; i'll manually write dp tables later
class Cow:
def __init__(self, score, ofFJ):
self.score = score
self.ofFJ = ofFJ
def __repr__(self):
return f"Cow({self.score}, {self.ofFJ})"
MOD = 1_000_000_009
def balancedArrangements(n):
@cache
def getBalancedArrangements(originaln, n, s):
"""
a balanced arrangement will correspond to a pair-off and will look like a list of bool; each bool will correspond to a cow; that bool will have index N iff the cow that it corresponds to has the (N+1)th lowest score in that pair-off, and that bool will be equal to whether the cow that that bool corresponds to is FJ's .
"""
if n == 0:
return [[True] * s]
else:
ans = []
if s < originaln:
for arrangement in getBalancedArrangements(originaln, n - 1, s + 1):
ans.append([False] + arrangement)
if s > 0:
for arrangement in getBalancedArrangements(originaln, n, s - 1):
ans.append([True] + arrangement)
return ans
return getBalancedArrangements(n, n, 0)
def main():
nFJCows, nFPCows, ncowsPerTeam = map(int, input().split())
cows = []
for ofFJ in True, False:
for score in map(int, input().split()):
cows.append(Cow(score, ofFJ))
cows.sort(key=lambda c: (c.score, int(c.ofFJ)))
print(cows)
winningTeamArrangements = balancedArrangements(ncowsPerTeam)
print(winningTeamArrangements)
seen_matches = set()
# obviously this comment will be replaced with the uncommented line @cache when nwaysToGetArrangement works and match -- a parameter that only exists in nwaysToGetArrangement for debugging purposes -- is no longer a parameter
def nwaysToGetArrangement(iarrangement, posInArrangement=0, icow=0, match=(), debug_print=False):
"""
where
iarrangement == index of the arrangement of cows (equivalent to subsequence of parentheses) are we trying to generate
posInArrangement answers "where in that arrangement are we currently?"
icow answers "which cow are we on currently?"
match answers "which cows are part of this way to pick winningTeamArrangements[iarrangement] from cows?"
debug_print is a variable that exists only so that i don't have to manually remove some debug print statements lol
"""
assert posInArrangement <= icow
if posInArrangement == len(winningTeamArrangements[iarrangement]):
if debug_print:
assert match not in seen_matches
print(f"\t{match}")
seen_matches.add(match)
return 1
elif icow == len(cows):
return int(posInArrangement == len(winningTeamArrangements[iarrangement]))
else:
ans = 0
if cows[icow].ofFJ == winningTeamArrangements[iarrangement][posInArrangement]:
ans += nwaysToGetArrangement(iarrangement, posInArrangement + 1, icow + 1, match + (icow,), debug_print)
if ans >= MOD:
ans -= MOD
ans += nwaysToGetArrangement(iarrangement, posInArrangement, icow + 1, match, debug_print)
if ans >= MOD:
ans -= MOD
return ans
ans = 0
for iarrangement in range(len(winningTeamArrangements)):
print(f"{iarrangement=} nwaysToGetArrangement(iarrangement)={nwaysToGetArrangement(iarrangement)})")
ans += nwaysToGetArrangement(iarrangement, debug_print=True)
if ans >= MOD:
ans -= MOD
print(ans)
if __name__ == '__main__':
main()