#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

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER
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1. Who invented Latin Squares and Latin-Greco Squares? What are they?
A1. Orthogonal Latin squares were studied in detail by Leonhard Euler, who took the two sets to 
	be S = {A, B, C, ...}, the first n upper-case letters from the Latin alphabet, and 
	T = {α , β, γ, ...}, the first n lower-case letters from the Greek alphabet—hence the 
	name Graeco-Latin square.

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2. What is nops(AllGraphs(20)) ?
A2. seq(nops(AllGraphs(i)), i = 1 .. 6);
                    1, 2, 8, 64, 1024, 32768

2^0, 2, 2^3, 2^6, 2^10, 2^15;
                    1, 2, 8, 64, 1024, 32768
# 2^0, 2, 2^3, 2^6, 2^10, 2^15, 2^21, 2^28, 2^36, 2^45, 2^55, 2^66, 2^78, 2^91, 2^105, 2^120, 2^136, 2^153, 2^171, 2^190, 2^210;

From this sequence we can see that nops(AllGraphs(20)) := 

2^210;
1645504557321206042154969182557350504982735865633579863348609024

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3. What is CC(G,38) ?
A3. 
G := RandGr(200, 50);
G := [{184}, {}, {121}, {}, {49}, {}, {65, 118}, {}, {}, {}, {}, 

  {}, {53, 186}, {}, {}, {}, {169}, {74}, {139}, {}, {}, {26}, 

  {}, {104}, {}, {22}, {}, {41}, {45}, {}, {121}, {156}, {}, {}, 

  {}, {}, {}, {}, {86, 197}, {}, {28}, {}, {}, {}, {29, 189}, 

  {53, 115}, {137}, {}, {5}, {121, 126}, {}, {149}, {13, 46}, {}, 

  {}, {173}, {}, {}, {158}, {}, {}, {}, {}, {}, {7}, {}, {}, {}, 

  {}, {86}, {176}, {}, {}, {18}, {}, {}, {200}, {}, {114, 144}, 

  {}, {}, {}, {154}, {}, {}, {39, 70}, {}, {200}, {}, {}, {}, 

  {94}, {}, {92}, {}, {135, 142}, {}, {}, {}, {}, {}, {}, {}, 

  {24}, {}, {118}, {}, {185}, {}, {}, {}, {}, {144}, {79}, {46}, 

  {}, {194}, {7, 106}, {}, {}, {3, 31, 50}, {}, {}, {}, {}, 

  {50, 157, 166}, {190}, {}, {}, {}, {}, {}, {175}, {184}, {96}, 

  {}, {47}, {}, {19}, {}, {184}, {96}, {}, {79, 113, 178}, {}, 

  {}, {}, {}, {52}, {}, {}, {197}, {}, {83}, {}, {32}, {126}, 

  {59}, {}, {}, {}, {}, {}, {}, {}, {126}, {}, {}, {17}, {}, {}, 

  {}, {56}, {}, {133}, {71}, {}, {144}, {185}, {}, {}, {}, {}, 

  {1, 134, 141}, {108, 179}, {13}, {}, {}, {45}, {127}, {}, {}, 

  {}, {117}, {}, {}, {39, 152}, {}, {}, {77, 88}]


CC(G, 38);
                              {38}

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4. NuConG(5) := [1,1,4,38,728] Is this sequence on OEIS? A-number?
A4. It is on OEIS. A-number := A001187

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5. G := [{4,8},{7},{5,9},{1,6},{3,7,10},{4,9},{2,5},{1},{3,6},{5}]
	Draw this graph manually.
A5. Attached as an image seperately

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6. [seq(nops(AllConGraphsK(n1,n1-1)),n1=1..6)] := [1,1,3,16,125,1296]
(i) Is this sequence on OEIS?
	Yes
(ii) A - number ?
	A-number := A000272
(iii) What is the number of (labelled) trees with 20 vertices?
	Follows the sequence with A-number := A001187

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7. [seq(nops(AllConGraphsK(n1,n1)),n1=3..6)] := [1, 15, 222, 3660]
	Is this sequence on OEIS? A-number and reason ?
A7. Yes it is on OEIS. A-number := A057500
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8. [seq(nops(AllConGraphsK(n1,n1+1)),n1=4..6)] := [6, 205, 5700]
	Is this sequence on OEIS? A-number?
A8. Yes it is on OEIS. A-number := A061540
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