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55 lines
1.0 KiB
Python
55 lines
1.0 KiB
Python
"""
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Counting Summations
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Problem 76
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It is possible to write five as a sum in exactly six different ways:
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4 + 1
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3 + 2
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3 + 1 + 1
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2 + 2 + 1
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2 + 1 + 1 + 1
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1 + 1 + 1 + 1 + 1
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How many different ways can one hundred be written as a sum of at least two
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positive integers?
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"""
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def partition(m):
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"""Returns the number of different ways one hundred can be written as a sum
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of at least two positive integers.
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>>> partition(100)
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190569291
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>>> partition(50)
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204225
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>>> partition(30)
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5603
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>>> partition(10)
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41
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>>> partition(5)
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6
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>>> partition(3)
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2
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>>> partition(2)
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1
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>>> partition(1)
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0
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"""
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memo = [[0 for _ in range(m)] for _ in range(m + 1)]
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for i in range(m + 1):
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memo[i][0] = 1
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for n in range(m + 1):
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for k in range(1, m):
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memo[n][k] += memo[n][k - 1]
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if n > k:
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memo[n][k] += memo[n - k - 1][k]
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return memo[m][m - 1] - 1
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if __name__ == "__main__":
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print(partition(int(str(input()).strip())))
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