#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. 6, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 7

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER


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Attendance Question 1:
Who invented Latin Squares and Latin-Greco Squares, and what are they?

Both were invented by Leonhard Euler.

Latin squares are nxn arrays filled with n different symbols such that each symbol can only occur in a row exactly once and a column exactly once.

Latin-Greco squares can be thought of as pairs of orthogonal Latin squares. Essentially, a Latin-Greco squares consists of an nxn array with each cell containing an ordered pair (s,t) where s in S and t in T and S and T each contain n symbols. The rule is that every row and every column must contain each element of S and each element of T exactly once, but no cells can contain the same ordered pair.



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Attendance Question 2:
What is nops(AllGraphs(20))?

2^binomial(20,2) = 2^190 = 1569275433846670190958947355801916604025588861116008628224

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Attendance Question 3:
What is CC(G,38)? 

{1, 38}

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Attendance Question 4:
Is this sequence in the OEIS? What is its A-number? (And description)
[1,1,4,38,728]

Yes, it is A1187. It is the number of connected labeled graps with n nodes.


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

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


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Attendance Question 6:
Is this sequence in the OEIS? What is the A-number? What is the number of (labeled) trees with 20 vertices?

[1,1,3,16,125,1296]

Yes, it is A272. It is the number of trees on n labeled nodes: n^(n-2).
20^18 = 262144000000000000000000.


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Attendance Question 6:
Is this in the OEIS? What is the A-number, reason?
[1,15,222,3660]

Yes, it is A57500 as the number of connected labeled graphs with n edges and n nodes.


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Attendance Question 7:
Is this in the OEIS? What is the A-number?
[6,205,5700]

Yes, it is A61540 as the number of connected labeled graphs with n nodes and n+1 edges.