#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

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Attendance Question 1:

Write down the set of integer partitions for 7.

{[7], [61], [52], [511], [43], [421], [4111], [331], [322], [3211], [31111], [2221], [22111], [211111], [1111111]}

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Attendance Question 2:

Write down the set of integer partitions of 7 into distinct parts and
write down the set of integer partitions of 7 into odd parts. Are they the same number?

Distinct parts:
{[7], [61], [52], [43], [421]}

Odd parts:
{[7], [511], [331], [31111], [1111111]}

They both have the same number of elements = 5.
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Attendance Question 3:

What is the A-number of this sequence?

A41

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Attendance Question 4:

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

George E. Andrews wrote The Theory of Partitions and his birthdate is 12/04/1938.

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Attendance Question 5:

What is the A number of this sequence? (integer partitions into distinct parts)

A9

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Attendance Question 6:

What is the A number of this sequence? (integer partitions into distinct parts where difference between succesive parts is at least 2)

A3114

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Attendance Question 7:

Whose theorem is it that these above two sequences are the same?

It seems that Leonard James Rogers and Srinivasa Ramanujan discovered and proved that the above two sequences are the same, although Issai Schur also independently discovered and proved the identities too.

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Attendance Question 8:

What is the A number of this sequence if it is in the OEIS?

A71734

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Attendance Question 9:

i) Does there exist a k between 0 and 6 such that
seq(pn(7*n+k/7, n=1..60); is an integer seequence?
What is the A-number if it exists?

Used {seq(pn(7*i+5) mod 7, i=1..60)}; for various k.

k = 5 works,
[seq(pn(7*n+5)/7, n = 1..20)];
[11, 70, 348, 1449, 5334, 17822, 55165, 160215, 441105, 1159752, 2929465, 7142275, 16873472, 38749850, 86737678, 189672868, 405991500, 852077072, 1756048833, 3558408287]

A71746


ii) Does there exist a k between 0 and 10 such that
seq(pn(11*n+k/11, n=1..60); is an integer seequence?
What is the A-number if it exists?

k = 6 works,
[seq(pn(11*n+6)/11, n = 1..20)];
[27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805845, 1117485621, 3366123200, 9767105406, 27398618368, 74534264393, 197147918679, 508189847045, 1279140518117, 3149375120229]

A76394

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Attendance Question 10:

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

It makes sense to watch lectures before recitation so that during recitation, if there are any questions or confusion about the material, that can be cleared up at recitation.