#ATTENDANCE QUIZ FOR LECTURE 24 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p24
#with an attachment called
#p24FirstLast.txt
#(e.g. p24DoronZeilberger.txt)

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


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 10

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

QUESTION #1:

What are all the integer partition of 7?

ANSWER:

{[7], [6,1], [5,2], [5,1,1], [4,3], [4, 2, 1], [4, 1, 1, 1], [3, 2, 1, 1], [3, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]}

QUESTION #2:

Write down the set of integer partitions for 7 into distinct parts and do the same for odd parts.

ANSWER:

Distinct Parts:
{[7], [6,1], [5,2], [4,3], [4,2,1]}

Odd Parts:
{[7], [5,1,1], [4,2,1], [3,1,1,1,1], [1,1,1,1,1,1,1]}

QUESTION #3:

What is the A-number of the sequence 1,2,3,5,7,11,15,22,30,42,56,77,101,...

ANSWER:

A000041
Description: a(n) is the number of partitions of n (the partition numbers).

QUESTION #4:

Who wrote the classic book on the theory of partitions? What is his birthdate?

ANSWER:

It was written by George Andrews, who was born on December 4th, 1938.

QUESTION #5:

What is the A-number of 1,1,2,2,3,4,5,6,8,10,12,15,18,22,27

ANSWER:

A000009
Description: Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.

QUESTION #6:

What is the A-number of 1,1,1,2,3,3,4,5,6,7,9,10,12,14,17,...

ANSWER:

A029036

QUESTION #7:

What theorem is it that the above two sequences are the same?

ANSWER:

It was Euler's theorem that Sum(p(n)*x^n, n=0..infty) = 1/((1-q)*(1-q^2)*...)

QUESTION #8:

What is the A-number of 6,27,98,315,913,2462,6237, 15035, 34705,...

ANSWER:

A071734
Description: a(n) = p(5n+4)/5 where p(k) denotes the k-th partition number.

QUESTION #9:

(i) Does there exist a k between 0 and 6 such that seq(pn(7*n+k)/7,n=1..20) is an integer sequence? What is the A-number?

(ii) Does there exist a k between 0 and 10 s.t seq(pn(11*n+k)/11,n=1..60); is an integer sequence? What is the A-number?

ANSWER:

(i) Yes, for k=5, we have 11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287.

Its A-number is A071746.

(ii) Yes, for k=6, we have 27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 3149375120229.

Its A-number is A076394.

QUESTION #10:

Why should Dr. Z from now on make you watch the lectures before the recitation?

ANSWER:

It would make sense if we were to first learn the programs used for the upcoming lecture in the recitation. That would save some time during the lecture and might motivate some of the theory. Also, more students would have a reason to watch the recitations if they needed it for the lecture.