#ATTENDANCE QUIZ FOR LECTURE 23 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p23
#with an attachment called
#p23FirstLast.txt
#(e.g. p23DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Dec. 4, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 3

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER

# Question 1: Use Lagrange Inversion Formula to find an explicit expression for the coeff
# of x^n in the power series of u(x) that satisfies the functional equation:
# (i) u(x) = x*(1+u(x))^2
# (ii) u(x) = x*(1+u(x))^3
# (iii) u(x) = x*(1+u(x))^k (for any positive integer k)

# Answer: 

# (i): 
# R(x) = 1/n * coeff of z^(n-1) in (1+z)^2 
# 1/n * coeff of z in ((1+z)^2)^n = 1/n * (2n C n-1) 

# (ii):
# R(x) = 1/n * coeff of z^(n-1) in (1+z)^3
# 1/n * coeff of z in (1+z)^3n = 1/n * (3n C n-1)

# (iii):

# R(x) = 1/n * coeff of z^(n-1) in (1+z)^k
# 1/n * coeff of z in (1+z)^kn = 1/n * (kn C n-1) 

1/n * coeff of z^n-1 (1+z)^2n

# ----------------------------------------------------------------------------------------------------

# Question 2: Consider the function given above:
# f(1) = 13, f(2) = 7, .... f(16 = 1
# (i) Draw this directed graph

# Find the doubly rooted tree that is outputted by the joyal bijection

# Check that it is as below:
# [[1,13],[1,16], [2,7], [2,8], [3,15],[4,10], [4,11], [5,13], [6,7], [6,12],[6,16], [9,11],[10,12],[11,14],[15,16]], [13,5]

# Answer:

# 1   2    3    4    5    6    7    8    9    10    11    12    13     14     15     16
# 13  7    15   10   13   16   6    2    11   12    4      6     5     11     16      1

# 	   14
#           |
#     9 -> 11 -> 4
#        	 |
#                10
#                |
#                12
# 	         |
# 8 -> 2 -> 7 -> 6 -> 16 -> 1 -> 13 <-> 5
#                     |
#                     15
#                     |
#                     3 

# Cycles: {5,13}
# pi(5) = 13, pi(13) = 5

#            	 8
#            	 |
#           	 2
#           	 |
#           	 7
#            	 |
#           	 6          14     
#                |          |     
# 3-> 15 -> 16 -> 12-> 10->4->11->9
#           |
#           1
#           |
#           *13 -> 5**

# ----------------------------------------------------------------------------------------------------
# Question 3: Finish this. Find the Pruffer Code (of length 11) of the original tree.

# [2, 4, 11, 8, 7, 6, 10, 3, 6, 3, 2]