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80 lines
2.6 KiB
Python
80 lines
2.6 KiB
Python
"""
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Print all the Catalan numbers from 0 to n, n being the user input.
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* The Catalan numbers are a sequence of positive integers that
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* appear in many counting problems in combinatorics [1]. Such
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* problems include counting [2]:
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* - The number of Dyck words of length 2n
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* - The number well-formed expressions with n pairs of parentheses
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* (e.g., `()()` is valid but `())(` is not)
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* - The number of different ways n + 1 factors can be completely
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* parenthesized (e.g., for n = 2, C(n) = 2 and (ab)c and a(bc)
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* are the two valid ways to parenthesize.
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* - The number of full binary trees with n + 1 leaves
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* A Catalan number satisfies the following recurrence relation
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* which we will use in this algorithm [1].
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* C(0) = C(1) = 1
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* C(n) = sum(C(i).C(n-i-1)), from i = 0 to n-1
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* In addition, the n-th Catalan number can be calculated using
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* the closed form formula below [1]:
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* C(n) = (1 / (n + 1)) * (2n choose n)
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* Sources:
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* [1] https://brilliant.org/wiki/catalan-numbers/
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* [2] https://en.wikipedia.org/wiki/Catalan_number
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"""
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def catalan_numbers(upper_limit: int) -> "list[int]":
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"""
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Return a list of the Catalan number sequence from 0 through `upper_limit`.
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>>> catalan_numbers(5)
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[1, 1, 2, 5, 14, 42]
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>>> catalan_numbers(2)
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[1, 1, 2]
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>>> catalan_numbers(-1)
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Traceback (most recent call last):
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ValueError: Limit for the Catalan sequence must be ≥ 0
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"""
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if upper_limit < 0:
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raise ValueError("Limit for the Catalan sequence must be ≥ 0")
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catalan_list = [0] * (upper_limit + 1)
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# Base case: C(0) = C(1) = 1
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catalan_list[0] = 1
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if upper_limit > 0:
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catalan_list[1] = 1
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# Recurrence relation: C(i) = sum(C(j).C(i-j-1)), from j = 0 to i
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for i in range(2, upper_limit + 1):
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for j in range(i):
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catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1]
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return catalan_list
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if __name__ == "__main__":
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print("\n********* Catalan Numbers Using Dynamic Programming ************\n")
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print("\n*** Enter -1 at any time to quit ***")
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print("\nEnter the upper limit (≥ 0) for the Catalan number sequence: ", end="")
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try:
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while True:
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N = int(input().strip())
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if N < 0:
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print("\n********* Goodbye!! ************")
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break
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else:
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print(f"The Catalan numbers from 0 through {N} are:")
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print(catalan_numbers(N))
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print("Try another upper limit for the sequence: ", end="")
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except (NameError, ValueError):
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print("\n********* Invalid input, goodbye! ************\n")
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import doctest
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doctest.testmod()
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