Every arc diagram ((io, jo), increasing order, i.e., io < i (in-1. Jn-1)) of n arcs on 2n points (0, 1,..., 2n-1) has n starting points it, which we will assume to be sorted in < ... < İn−1 · For a general arc diagram, the end points j must satisfy i
Every arc diagram ((io, jo), increasing order, i.e., io < i (in-1. Jn-1)) of n arcs on 2n points (0, 1,..., 2n-1) has n starting points it, which we will assume to be sorted in < ... < İn−1 · For a general arc diagram, the end points j must satisfy i
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Every arc diagram ((io, jo),
increasing order, i.e., io < i
(in-1. Jn-1)) of n arcs on 2n points (0, 1,..., 2n-1) has n starting points it, which we will assume to be sorted in
< ... < İn−1 ·
For a general arc diagram, the end points j must satisfy i<jk, but otherwise can be chosen freely. Therefore, a brute-force way of finding all arc diagrams
on arcs is to consider all (2") combinations of starting points, and for each of these combinations to consider all »! permutations of the corresponding end
points. If all starting and ending points satisfy i <jk then this results in an arc diagram.
Use this method to write a function find_all_arc_diagrams that takes an integer n and returns a list of all arc diagrams on n arcs.
Test your code by printing out a list of all arc diagrams for n = 3 and by verifying that the total number of obtained diagrams is (2n- 1)!! for a couple of
(small) values of n.
You might find it helpful to loop through permutations of a list using itertools.permutation (imported above) by adopting code similar to
perms-permutations (list_of_items)
for p in perms:
and similarly for combinations using itertools.combination (imported above).
Transcribed Image Text:You must not import any other modules.
#DO NOT CHANGE THE CONTENT OF THIS CODE BOX
import matplotlib.pyplot as plt
import numpy as np
import matplotlib.patches as patches
from timeit import timeit
import seaborn as sns
from itertools import permutations, combinations
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