#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:

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

Q1. 
    Use lagrange inversion formula to find an explicit expression for the coefficients 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))^2
    
    (iii).
        u(x) = x * (1 + u(x))^k

A1. 
    (i). 
        phi(z) = (1 + z)^2
        coeff = 1/n * coeff of z^(n-1)

    (ii). 
        phi(z) = (1 + z)^3
        coeff = 1/n * coeff of z^(n-1)
    
    (iii). 
        phi(z) = (1 + z)^k 


Q2. 
    Consider the function given above 
    
    [01, 2, 03, 04, 05, 06, 7, 8, 09, 10,11,12,13, 14, 15,16]
    [13, 7, 15, 10, 13, 16, 6, 2, 11, 12, 4, 6, 5, 11, 16, 1]

    f(1) = 13, f(2) = 7, ... f(16) = 1

    (i). 
        Draw this directed graph 

    (ii). 
        Find the doubly-rooted tree that is outputted by the Joyal Bijection by hand. 
        Check that it is below

A2. 
    (i). 
                  1->13<->5
                  ^
                  |
      8->2->7->6->16<-15<-3
               ^
               |
 9->11->4->10->12
    ^
    |
    14
        
    (ii). 
        Cycles: 
        (13, 5) [only one]

        stem: 
        *13->5**

        restoration: 

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

Q3. 
    Finish this, find the pruffer code. 

A3. 
    current state: [2, 4, 11]

                13
                |
                2
                |
    9->8->7->6->3->10->11
             |
             12
    
    next smallest leaf: 9, neighbor: 8
    a4 = 8

                13
                |
                2
                |
       8->7->6->3->10->11
             |
             12    

    next smallest leaf: 8, neighbor: 7
    a5 = 7
                13
                |
                2
                |
          7->6->3->10->11
             |
             12       

    a6 = 6
                13
                |
                2
                |
             6->3->10->11
             |
             12           

    a7 = 10
                13
                |
                2
                |
             6->3->10
             |
             12           

    a8 = 3
                13
                |
                2
                |
             6->3
             |
             12               

    a9 = 6
                13
                |
                2
                |
             6->3

    a10 = 3
                13
                |
                2
                |
                3
    a11 = 2
                13
                |
                2