#ATTENDANCE QUIZ FOR LECTURE 18 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p18
#with an attachment called
#p18FirstLast.txt
#(e.g. p18DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 10, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE:8

#Q1.
#Who invented latin squares and latin-greco squares?
#what are they?

#A:
#Latin squares and latine-greco squares are invented by Euler
#Latin square is  n Ąż n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column
#

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#Q2.
#what is nops(AllGraphs(20))

#A: nops(AllGraphs) follows the occurence

seq(nops(AllGraphs(n)), n = 1 .. 5);
                       1,  2,   8,   64,  1024  
#this follows the    2^0  2^1  2^3   2^6   2^10
#which we can see that the powers are binomial(n, 2)(=n(n-1)/2)

#so nops(AllGraphs(n)) has an explicit formula of 2^(n(n-1)/2)

#then we now know that

#nops(AllGraphs(20)) = 2^(20*19/2) = 

#1569275433846670190958947355801916604025588861116008628224

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#Q3.
#What is CC(G,38)?
#Just following the logic,
#Since we obtained CC(G,1) = {1,38}, we can easily predict that it is closed loop between 1 <-> 38
#So CC(G,38) should also be equal to CC(G,1) = {1,38}

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#Q4.
#the sequence is NuConG(5)
#[1,1,4,38,728]
#Is this in OEIS and what is its A-number?

#A: A001187, It is called Number of connected labeled graphs with n nodes.

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#Q5.
#G=[{4,8},{7},{5,9},{1,6},{3,7,10},{4,9},{2,5},{1},{3,6},{5}]
#Draw this graph manually

#A: 1 - 4 - 6 - 9 - 3 - 5 - 7 - 2
#   |                   |
#   8                   10
#

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#Q6.
#1. IS this Seqence in the OEIS?
#2. What is the A-#
#3. What is the number of (labeled) trees with 20 vertices?

#A: A000272, Number of trees on n labeled nodes: n^(n-2) with a(0)=1.
#Since it takes n-1 vertices,
#n=21,
#21^(21-2) = 13248496640331026125580781 = the number of (labeled) trees with 20 vertices?

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#Q7. 

#seq(nops(AllConGraphsK(n1, n1)), n1 = 3 .. 6);
#                        1, 15, 222, 3660
#IS this in OEIS?

#A: Yes , A057500 ( Number of connected labeled graphs with n edges and n nodes.)

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#Q8. 
#seq(nops(AllConGraphsK(n1, n1 + 1)), n1 = 4 .. 6);
#                          6, 205, 5700

#is this in the OIES?

#A: A061540, Number of connected labeled graphs with n nodes and n+1 edges.