diff --git a/dynamic_programming/catalan_numbers.py b/dynamic_programming/catalan_numbers.py new file mode 100644 index 000000000000..7b74f2763d43 --- /dev/null +++ b/dynamic_programming/catalan_numbers.py @@ -0,0 +1,79 @@ +""" +Print all the Catalan numbers from 0 to n, n being the user input. + + * The Catalan numbers are a sequence of positive integers that + * appear in many counting problems in combinatorics [1]. Such + * problems include counting [2]: + * - The number of Dyck words of length 2n + * - The number well-formed expressions with n pairs of parentheses + * (e.g., `()()` is valid but `())(` is not) + * - The number of different ways n + 1 factors can be completely + * parenthesized (e.g., for n = 2, C(n) = 2 and (ab)c and a(bc) + * are the two valid ways to parenthesize. + * - The number of full binary trees with n + 1 leaves + + * A Catalan number satisfies the following recurrence relation + * which we will use in this algorithm [1]. + * C(0) = C(1) = 1 + * C(n) = sum(C(i).C(n-i-1)), from i = 0 to n-1 + + * In addition, the n-th Catalan number can be calculated using + * the closed form formula below [1]: + * C(n) = (1 / (n + 1)) * (2n choose n) + + * Sources: + * [1] https://brilliant.org/wiki/catalan-numbers/ + * [2] https://en.wikipedia.org/wiki/Catalan_number +""" + + +def catalan_numbers(upper_limit: int) -> "list[int]": + """ + Return a list of the Catalan number sequence from 0 through `upper_limit`. + + >>> catalan_numbers(5) + [1, 1, 2, 5, 14, 42] + >>> catalan_numbers(2) + [1, 1, 2] + >>> catalan_numbers(-1) + Traceback (most recent call last): + ValueError: Limit for the Catalan sequence must be ≥ 0 + """ + if upper_limit < 0: + raise ValueError("Limit for the Catalan sequence must be ≥ 0") + + catalan_list = [0] * (upper_limit + 1) + + # Base case: C(0) = C(1) = 1 + catalan_list[0] = 1 + if upper_limit > 0: + catalan_list[1] = 1 + + # Recurrence relation: C(i) = sum(C(j).C(i-j-1)), from j = 0 to i + for i in range(2, upper_limit + 1): + for j in range(i): + catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1] + + return catalan_list + + +if __name__ == "__main__": + print("\n********* Catalan Numbers Using Dynamic Programming ************\n") + print("\n*** Enter -1 at any time to quit ***") + print("\nEnter the upper limit (≥ 0) for the Catalan number sequence: ", end="") + try: + while True: + N = int(input().strip()) + if N < 0: + print("\n********* Goodbye!! ************") + break + else: + print(f"The Catalan numbers from 0 through {N} are:") + print(catalan_numbers(N)) + print("Try another upper limit for the sequence: ", end="") + except (NameError, ValueError): + print("\n********* Invalid input, goodbye! ************\n") + + import doctest + + doctest.testmod()