#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


Question 1:

Who invented Latin squares and Latin-Greco squares? What are they?


Answer 1:

Leonhard Euler coined the term "Latin squares"

A Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column

ex:     A	B	C

	C	A	B

	B	C	A


Two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A pair of orthogonal Latin squares are traditionally called Greco-Latin squares. 




Question 2:

What is nops(AllGraphs(20)) ?


Answer 2:

nops(AllGraphs(20)) is 2^(20C2). There are binomial(20,2) possible edges in the graph and we can choose to have an edge or not. Hence, 2^(20C2).


Question 3:

Whats is CC(6,38)?


Answer 3:

The only neighbors of verdict 38 is {1} so the connected component is {1,38}.


Question 4:

[1,1,4,38,728] What is the A number and description?


Answer 4:

The a number is A001187. 

Description: Number of connected labeled graphs with n nodes.



Question 5:

Draw the tree. 


Answer 5:

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

     	        10




Question 6:

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

Is this sequence in the OEIS?
What is the A number and description?
What is the number of labeled trees with 0 vertices?

Answer 6:

Yes the sequence is in the OEIS.
The a number is A000272
Description: Number of trees on n labeled nodes: n^(n-2) with a(0)=1

The number of labeled trees with twenty vertices is 20^(18) = 2.62144e+23


Question 7:

n vertices with n edges
Almost trees [1,15,222,3660]
Is this in the OEIS?
What is the A number, description?


Answer 7:

Yes this is in the OEIS. 
A057500 
Description: Number of connected labeled graphs with n edges and n nodes.



Question 8:

N vertices with n+1 edges
Almost Almost trees [6,205,5700]
Is it in the OEIS? What is the A number? Description?


Answer 8:

Yes it is in the OEIS - A061540
Description : Number of connected labeled graphs with n nodes and n+1 edges.