#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 the Lagrange inversion formula to find an explicit expression for the coeff of x^n in the power series of u(x) that satisfies 

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

Answer 1: for i I got the beautiful catalan numbers
k = 2 #binomial(2n,n)/(n+1)
k = 3 #binomial(3*n,n)/(2*n+1) 
for k binomial(k*n, n)/(k-1)*n+1)

Question 2:
Consider the function 
[13, 7, 15, 10, 13, 16, 6, 2, 11, 12, 4, 6, 5, 11, 16, 1]
i) Draw this directed graph
Find the doubly rooted tree that is outputted by the joyal bijection

Answer 2:
i) 1->13-5
{{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]


Question 3: Find the pruffer code of (lenghth 11) of the original tree

Answer 3:
[3, 12, 8, 10, 3, 1, 1, 13, 3, 5, 5]