#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

1) Use lagrange inversion formula to find an explicit expression for the 
coefficient of x^n in the power series of u(x) that satisfies the functional equations

The coeff of x^n in u(x)=(1/n)*COEFF of z^(n-1) in PHI(z)^n where PHI(z)=exp(z)

(i) u(x)=x*(1+u(x))^2

(ii) u(x)=x*(1+u(x))^3

(iii) For any k, Positive integer

Sorry, I don't know how to do this. 

2) 
L:=[13,7,15,10,13,16,7,2,11,12,4,6,5,11,16,1]
Consider the function given above: f(1)=13, f(2)=7, ... f(16)=1
1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16
13 7  15 10 13 16 7  2  11 12  4   6   5   11  16   1

(i) Draw this directed graph

8->2->7<->

    14            3->15
    |                 |
    +                 +
9->11->4->10->12->6->16->1->13<->5

(ii) Find the doubly-rooted tree that is outputted by 
the joyal bijection by hand

Cycles: (7) (13,5)

pi(7)=7
pi(13)=5
pi(5)=13

5   7  13 
13  7  5

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


(ii) Check that it as below

Joyal(L);
{{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]

3) Finish this. Find the pruffer code of length 11 for the original tree

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

a1=2

New tree

         13
         |
         2
         |
9->8->7->6->3->10->11->4
         |             |
         12            5

a2=4 code so far [2,4]

         13
         |
         2
         |
9->8->7->6->3->10->11->4
         |             
         12

a3=11 code so far [2,4,11]          

         13
         |
         2
         |
9->8->7->6->3->10->11
         |             
         12  

a4=8 code so far [2,4,11,8]

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

a5=7 code so far [2,4,11,8,7]   

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

a6=6 code so far [2,4,11,8,7,6] 

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

a7=10 code so far [2,4,11,8,7,6,10] 

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

a8=3. code so far [2,4,11,8,7,6,10,3] 

  13
   |
   2
   |
   6->3
   |             
   12  

a9=6. code so far [2,4,11,8,7,6,10,3,6] 

  13
   |
   2
   |
   6
   |             
   12 

a10=12 code so far [2,4,11,8,7,6,10,3,6,12] 

  13
   |
   2
   |
   6

a11=2 code so far [2,4,11,8,7,6,10,3,6,12,2] 

  13
   |
   2

Pruffer Code: [2,4,11,8,7,6,10,3,6,12,2]