def factorial(n): """This function calculates the factorial of a non-negative integer n."""" result = 1 for i in range(1, n + 1): result = i return result def get_all_matchings(n): """This function generates all matchings of size n recursively. Args: n: A non-negative integer representing the number of arcs. Returns: A list of all possible matchings for n arcs. IIIIII if n 0: return [ [ ] ] elif n == 1: else: return [[(0, 1)]] matchings_n_minus_1 matchings n = [] = get_all_matchings (n 1) for matching in matchings_n_minus_1: for i in range(len(matching) + 1): new_matching = matching [i] + [(0, n)] + [(x[0] + 1, x[1] + 1) for x in matching [i:]] matchings_n.append(new_matching) return matchings_n for n in range(8): # Test cases num_matchings = len (get_all_matchings(n)) expected_matchings = factorial (2*n) // factorial(n) print(f"n={n}, Expected number of matchings: {expected_matchings}, Actual number of matchings: {num_matchings}") = 6, compute the number C₁₂ of arc diagrams with k crossings and I nestings for all possible values of k and 7. Store these counting numbers Ckl in a numpy array of integer data type and appropriate shape, meaning that the number of rows and columns should be aligned with the maximum numbers of crossings and nestings, respectively. You should get a symmetric array. Print this array. Furthermore, also print an array showing the total numbers of crossings for every k and, analogously, an array showing the total number of nestings for every 1.
def factorial(n): """This function calculates the factorial of a non-negative integer n."""" result = 1 for i in range(1, n + 1): result = i return result def get_all_matchings(n): """This function generates all matchings of size n recursively. Args: n: A non-negative integer representing the number of arcs. Returns: A list of all possible matchings for n arcs. IIIIII if n 0: return [ [ ] ] elif n == 1: else: return [[(0, 1)]] matchings_n_minus_1 matchings n = [] = get_all_matchings (n 1) for matching in matchings_n_minus_1: for i in range(len(matching) + 1): new_matching = matching [i] + [(0, n)] + [(x[0] + 1, x[1] + 1) for x in matching [i:]] matchings_n.append(new_matching) return matchings_n for n in range(8): # Test cases num_matchings = len (get_all_matchings(n)) expected_matchings = factorial (2*n) // factorial(n) print(f"n={n}, Expected number of matchings: {expected_matchings}, Actual number of matchings: {num_matchings}") = 6, compute the number C₁₂ of arc diagrams with k crossings and I nestings for all possible values of k and 7. Store these counting numbers Ckl in a numpy array of integer data type and appropriate shape, meaning that the number of rows and columns should be aligned with the maximum numbers of crossings and nestings, respectively. You should get a symmetric array. Print this array. Furthermore, also print an array showing the total numbers of crossings for every k and, analogously, an array showing the total number of nestings for every 1.
Programming Logic & Design Comprehensive
9th Edition
ISBN:9781337669405
Author:FARRELL
Publisher:FARRELL
Chapter6: Arrays
Section: Chapter Questions
Problem 19RQ
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