#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

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Attendance Question 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:

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

(1/n) * Coeff of z^(n-1) in Phi(z)^n
Phi(z)^n = (1+z)^(2n) = Sum(binomial(2n,i)*z^i, i = 0..2n)
Coeff of z^(n-1) in Phi(z)^n = binomial(2n,n-1) = (2n)!/((n-1)!*(n+1)!)

So, the coefficient of x^n in the power series of u(x) is (2n)!/(n!*(n+1)!).

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

1/n * binomial(3n,n-1) = (3n)! / (n!*(2n+1)!)


iii) For any k, pos. integer, u(x) = x*(1+u(x))^k

(1/n) * Coeff of z^(n-1) in Phi(z)^n

1/n * binomial(kn,n-1) = (kn)! / (n!*((k-1)*n+1)!)

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Attendance Question 2:

Consider the function given above:

f(1)=13, f(2)=7, f(3)=15, f(4)=10, f(5)=13, f(6)=16, f(7)=6, f(8)=2, f(9)=11, f(10)=12, f(11)=4, f(12)=6, f(13)=5, f(14)=11, f(15)=16, f(16)=1.

i) Draw this directed graph

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


Find the Doubly-Rooted tree that is outputted by the Joyal bijection:

The only cycle is (5,13)

5  13

13  5



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

Check that it is as below
Joyal([]);

{{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]

13 and 5 are the roots as drawn above. All of the edges are listed as in the tree.


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Attendance Question 3:

Finish this and find Pruffer Code (of length 11) of the original tree

[2,4,11]
Leaf: 9  [2,4,11,8]
Leaf: 8  [2,4,11,8,7]
Leaf: 7  [2,4,11,8,7,6]
Leaf: 11 [2,4,11,8,7,6,10]
Leaf: 10 [2,4,11,8,7,6,10,3]
Leaf: 12 [2,4,11,8,7,6,10,3,6]
Leaf: 6  [2,4,11,8,7,6,10,3,6,3]
Leaf: 3  [2,4,11,8,7,6,10,3,6,3,2]