#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p24
#with an attachment called
#p24FirstLast.txt
#(e.g. p24DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Dec. 8, 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 to find an explicit expression for the coeff of x^n in the power series of u(x) that satisfies the functional equations

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

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

(iii) u(x)= x*(1+u(x))^k


Answer 1:

(i) 
Using Lagrange's formula, the coefficient of x^n in u(x) is

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

(ii)
Using Lagrange's formula, the coefficient of x^n in u(x) is

(1/n) * coeff of z^(n-1) in (1+z)^(3n)


(iii)
Using Lagrange's formula, the coefficient of x^n in u(x) is

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



Question 2:

RandF(16)

Consider the function given above (in lecture)

(i) draw the directed graph
(ii)find the doubly rooted tree output by the royal bijection
(iii) Check on paper that it is as below
Joyal(RandF(16)) 

Answer 2:

#My drawing agrees with Joyal(RandF(16));


Question 3:

Finish this. Find the Pruffer code (for length 11 of original tree) (tree in lecture)

Answer 3:

The Pruffer code of the tree in the lecture is

[2,4,11,8,7,6,10,3,6,3,2,13]