#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:

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


QUESTION #1:

Who invented latin squares and latin-greco squares? What are they?

ANSWER:

Mathematician Choi Seok-jeong was the first to propose it in 1700. Latin squares are n x n arrays whose rows have n distinct elements and whose columns also have n distinct elements. The same n elements.

Mathematician Jacques Ozanam proposed latin-greco square in 1725. It is a pair of latin squares over two sets S and T of size n, where each cell contains a pair (s,t), arranged in a n x n grid. It has the property that each row contains each element of S and T exactly once and no two cells contain the same pair.
 

QUESTION #2:

What is nops(AllGraphs(20));? 

ANSWER:

Running seq(nops(AllGraphs(i)), i = 1 .. 5) we get:

1, 2, 8, 64, 1024,..

So the sequence is defined by a(n) = 2^(n*(n-1)/2).
Plugging in 20 we get, a(20) = 2^190 = 1.569275434 x 10^57.

QUESTION #3:

What is CC(G,38)?

ANSWER:

For the given G, the connected component of 38 is {1}.


QUESTION #4:

Is [1,1,4,38,728] in the OEIS? What is its A-number?

ANSWER:

Yes, it's A-number is A001187 and the description is: Number of connected labeled graphs with n nodes.

QUESTION #5:

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

ANSWER:

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

QUESTION #6:

(i) Is [1,1,3,16,125,1296] in the OEIS?
(ii) What is the A-number?
(iii) What is the number of (labeled) trees with 20 vertices?

ANSWER:

(i) Yes.
(ii) Its A-number is A000272: Number of trees on n labeled nodes: n^(n-2) with a(0)=1.
(iii) Using the equaition from part (ii), we have 20^18 number of trees with 20 vertices: 2.262144 x 10^23.


QUESTION #7:

Is [1,15,222,3660] in the OEIS? What is the A-number and reason?

ANSWER:

Yes, its A-number is A057500, description: Number of connected labeled graphs with n edges and n nodes.


QUESTION #8:

Is [6,205,5700] in the OEIS? What is the A-number?

ANSWER:

Yes, its A-number is A061540. It represents the number of connected labeled graphs with n nodes and n+1 edges.