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# 3539. Find Sum of Array Product of Magical Sequences
"""
you are given two integers 'm' and 'k', and an integer array 'nums'. a
sequence of integers 'seq' is called magical if 'seq' has a size of 'm', '0
<= seq[i] < nums.len', and the binary representation of '2^seq[0] + 2^seq[1]
+ ...' has 'k' set bits. the array product of this sequence is defined as
'prod(seq) = (nums[seq[0]] nums[seq[1]] ...)'. return the sum of the
array products for all valid magical sequences. since the answer may be very
large, return it modulo 10^9+7. a set bit refers to a bit in the binary
represenation of a number that has a value of 1
"""
class Solution(object):
def magicalSum(self, m, k, nums):
"""
:type m: int
:type k: int
:type nums: List[int]
:rtype: int
"""
mod = 10**9 + 7
n = len(nums)
powtab = [[1] * (m + 1) for _ in range(n)]
for i in range(n):
for c in range(1, m + 1):
powtab[i][c] = powtab[i][c - 1] * nums[i] % mod
comb = [[0] * (m + 1) for _ in range(m + 1)]
for i in range(m + 1):
comb[i][0] = 1
for j in range(1, i + 1):
comb[i][j] = (comb[i - 1][j - 1] + comb[i - 1][j]) % mod
dp = [[[0] * (k + 1) for _ in range(m + 1)] for __ in range(m + 1)]
dp[m][0][0] = 1
for i in range(n):
next_dp = [[[0] * (k + 1) for _ in range(m + 1)] for __ in range(m + 1)]
pow_i = powtab[i]
for rem in range(m + 1):
for carry in range(m + 1):
row = dp[rem][carry]
if not any(row_elt for row_elt in row):
continue
for ones in range(k + 1):
base_val = row[ones]
if base_val == 0:
continue
for c in range(rem + 1):
t = c + carry
bit = t & 1
ones2 = ones + bit
if ones2 > k:
continue
carry2 = t >> 1
rem2 = rem - c
add = base_val
add = add * comb[rem][c] % mod
add = add * pow_i[c] % mod
next_dp[rem2][carry2][ones2] = (
next_dp[rem2][carry2][ones2] + add
) % mod
dp = next_dp
ans = 0
for carry in range(m + 1):
extra = bin(carry).count("1")
need = k - extra
if 0 <= need <= k:
ans = (ans + dp[0][carry][need]) % mod
return ans % mod
if __name__ == "__main__":
obj = Solution()
print(obj.magicalSum(m=5, k=5, nums=[1, 10, 100, 10000, 1000000]))
print(obj.magicalSum(m=2, k=2, nums=[5, 4, 3, 2, 1]))
print(obj.magicalSum(m=1, k=1, nums=[28]))
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