def lexi_after(alpha, n):
count = 0
for i in range(alpha):
for j in range(i + 1, alpha):
for a, b, c in pick_n((i, j - i - 1, alpha - j - 1), n - 2):
count += nCr(i, a) * nCr(j - i - 1, b) * 2**b * nCr(
alpha - j - 1, c)
return count
def main():
size = 250250
div = 250
mod = 10 ** 16
remainder = [0] * div
cache = [[0] * div for _ in range(div)]
# cache = np.zeros((div, div), dtype=np.int64)
for i in range(1, size + 1):
remainder[0 if i % 10 == 0 else pow(i, i, 250)] += 1
# print(remainder)
cache[0][0] = pow(2, remainder[0], mod) - 1
for rem in range(1, div):
print(rem)
change = [0] * div
for c in range(remainder[rem] + 1):
change[rem * c % div] = (change[rem * c % div] + nCr(remainder[rem], c)) % mod
# print(change)
for index, i in enumerate(change):
for j in range(div):
new = cache[rem][(index + j) % div] + i * cache[rem - 1][j]
cache[rem][(index + j) % div] = new % mod
# print(rem, cache[rem])
print(cache[-1][0] - 1)
def main():
size = 250250
div = 250
mod = 10 ** 16
remainder = [0] * div
cache = [[0] * div for _ in range(div)]
# cache = np.zeros((div, div), dtype=np.int64)
for i in range(1, size + 1):
remainder[0 if i % 10 == 0 else pow(i, i, 250)] += 1
# print(remainder)
cache[0][0] = pow(2, remainder[0], mod) - 1
for rem in range(1, div):
print(rem)
change = [0] * div
for c in range(remainder[rem]+1):
change[rem * c % div] = (change[rem * c % div] + nCr(remainder[rem], c)) % mod
# print(change)
for index, i in enumerate(change):
for j in range(div):
new = cache [rem] [(index+j)%div] + i * cache [rem-1] [j]
cache [rem] [(index+j)%div] = new % mod
# print(rem, cache[rem])
print(cache[-1][0] - 1)
def partition_maxlen(n, maxlen):
total = 0
for length, count in partition_length(n).items():
if length > maxlen:
continue
total += count * nCr(maxlen, length)
return total
def main():
n = 12
count = 0
for i in range(2, n//2+1):
this_count = check(i)
this_count *= nCr(n, 2*i)
count += this_count
print(count)
def main():
n = 12
count = 0
for i in range(2, n // 2 + 1):
this_count = check(i)
this_count *= nCr(n, 2 * i)
count += this_count
print(count)
def main():
"""
def get_digits(n):
return list(map(int, str(n)))
same = 0
increasing = 0
decreasing = 0
for i in range(1, 10 ** 6):
all_digits = get_digits(i)
if all([j == all_digits[0] for j in all_digits]):
same += 1
continue
all_increasing = all(all_digits[j-1] <= all_digits[j] for j in range(1, len(all_digits)))
if all_increasing:
increasing += 1
continue
all_decreasing = all(all_digits[j-1] >= all_digits[j] for j in range(1, len(all_digits)))
if all_decreasing:
decreasing += 1
count = same + increasing + decreasing
print(same, increasing, decreasing)
print(count, 10**6-1 - count)
# experiments finished
"""
digits = 100
increase_cache = np.zeros((10, digits+1), dtype=np.int64) # 1~9 increase_cache[a, b] --> all digits <= a; b digits
increase_cache[1:, 0] = [1] * 9
# print(increase_cache)
for i in range(1, digits+1):
for j in range(1, 10):
increase_cache[j, i] = increase_cache[j-1, i] + increase_cache[j, i-1]
increase = sum(increase_cache[9, i] for i in range(1, digits+1))
# print(increase_cache)
print(increase)
decrease_cache = np.zeros((11, digits+1), dtype=np.int64) # 9~0 decrease_cache[a, b] --> all digits >= a; b digits
decrease_cache[1:10, 0] = [1] * 9
# print(decrease_cache)
for i in range(1, digits+1):
for j in range(9, -1, -1):
decrease_cache[j, i] = decrease_cache[j+1, i] + decrease_cache[j, i-1]
decrease = sum(decrease_cache[0, i] for i in range(1, digits+1))
# print(decrease_cache)
print(decrease)
result = increase + decrease - 9 * digits # 11111111... ~ 99999999...
print(result)
print("\nAnother extremely elegant solution:")
count = 0
for i in range(1, 101):
count += nCr(i+8, i) # increase up to 8 times
count += nCr(i+9, i) # decrease up to 9 times
count -= 10
print(count)
def main():
"""
def get_digits(n):
return list(map(int, str(n)))
same = 0
increasing = 0
decreasing = 0
for i in range(1, 10 ** 6):
all_digits = get_digits(i)
if all([j == all_digits[0] for j in all_digits]):
same += 1
continue
all_increasing = all(all_digits[j-1] <= all_digits[j] for j in range(1, len(all_digits)))
if all_increasing:
increasing += 1
continue
all_decreasing = all(all_digits[j-1] >= all_digits[j] for j in range(1, len(all_digits)))
if all_decreasing:
decreasing += 1
count = same + increasing + decreasing
print(same, increasing, decreasing)
print(count, 10**6-1 - count)
# experiments finished
"""
digits = 100
increase_cache = np.zeros(
(10, digits + 1), dtype=np.int64
) # 1~9 increase_cache[a, b] --> all digits <= a; b digits
increase_cache[1:, 0] = [1] * 9
# print(increase_cache)
for i in range(1, digits + 1):
for j in range(1, 10):
increase_cache[j, i] = increase_cache[j - 1,
i] + increase_cache[j, i - 1]
increase = sum(increase_cache[9, i] for i in range(1, digits + 1))
# print(increase_cache)
print(increase)
decrease_cache = np.zeros(
(11, digits + 1), dtype=np.int64
) # 9~0 decrease_cache[a, b] --> all digits >= a; b digits
decrease_cache[1:10, 0] = [1] * 9
# print(decrease_cache)
for i in range(1, digits + 1):
for j in range(9, -1, -1):
decrease_cache[j, i] = decrease_cache[j + 1,
i] + decrease_cache[j, i - 1]
decrease = sum(decrease_cache[0, i] for i in range(1, digits + 1))
# print(decrease_cache)
print(decrease)
result = increase + decrease - 9 * digits # 11111111... ~ 99999999...
print(result)
print("\nAnother extremely elegant solution:")
count = 0
for i in range(1, 101):
count += nCr(i + 8, i) # increase up to 8 times
count += nCr(i + 9, i) # decrease up to 9 times
count -= 10
print(count)