#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




#ATTENDANCE Q. #1 for LECTURE 23
#use lagrange inversion formula to find an excplicit 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 i)) u(x)=x*(1+u(x))^3 iii) for any k, u(x)=x*(1+u(x))^k

#ANSWER to Q. #1:	

# i) PHI(z)^n = (1+z)^(2*n), and so 1/n*coeff((1+z)^(2*n),z,n-1) = 1,2,5,14,42 (n=1..5) (Catalan numbers)  -->  the coeff of x^n is binomial(2n,n)/(n+1)
# ii) PHI(z)^n = (1+z)^(3*n), and so 1/n*coeff((1+z)^(3*n),z,n-1) = 1,3,12,55,273 (n=1..5) (enumeration of ternary trees and noncrossing trees)
#	From OEIS: A001764, so we have --> the coeff of x^n is binomial(3n,n)/(2n+1)
# iii) PHI(z)^n = (1+z)^(k*n), and so 1/n*coeff((1+z)^(k*n),z,n-1) [Following the pattern] --> the coeff of x^n is binomial(k*n,n)/((k-1)*n+1)




#ATTENDANCE Q. #2 for LECTURE 23
#consider the function given by [13,7,15,10,13,16,6,2,11,12,4,6,5,11,16,1]
#(i) Draw this directed graph
#(ii) Find the double-rooted tree that is output by the joyal bijection 
# Check that it as below
# {{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]

#ANSWER to Q. #2:
(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)

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

(ii)

#Only one cycle, it is (5,13)
#Two-Line notation:
5 13
13 5
#One line notation: 13 5

We have the double-rooted tree:

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





#ATTENDANCE Q. #3 for LECTURE 23
#Find the pruffer code (of len 11) for the tree. CODE SO FAR [2,4,11]

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

#ANSWER to Q. #3:

#Smallest leaf = 9; neighbor = 8; code so far [2,4,11,8]
#Smallest leaf = 8; neighbor = 7; code so far [2,4,11,8,7]
#Smallest leaf = 7; neighbor = 6; code so far [2,4,11,8,7,6]

#Tree as of now:

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

#Smallest leaf = 10; neighbor = 3; code so far [2,4,11,8,7,6,3]
#Smallest leaf = 12; neighbor = 6; code so far [2,4,11,8,7,6,3,6]
#Smallest leaf = 6; neighbor = 3; code so far [2,4,11,8,7,6,3,6,3]
#Smallest leaf = 3; neighbor = 2; code so far [2,4,11,8,7,6,3,6,3,2]
#Last smallest leaf = 2; neighbor = 13; code is [2,4,11,8,7,6,3,6,3,2,13]