#ATTENDANCE QUIZ FOR LECTURE 20 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p20
#with an attachment called
#p20FirstLast.txt
#(e.g. p19DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 17, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: ????????

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

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Attendance Question 1:

When was Rabbi Levi Ben Gerson (Gersonides) born? When did he die?

He was born in 1288 and died in 1344.

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Attendance Question 2:

When was E. T. Bell born? When did he die?
What is he famous for?

Eric Temple Bell was born on February 7, 1883 and died on December 21, 1960. He was famous for writing a book of biographical essays called "Men of Mathematics" (although he wanted the title to be "The Lives of Mathematicians").

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Attendance Question 3:

What was the pen name of E. T. Bell?

His pen name was John Taine.

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Attendance Question 4:

How many set partitions of a 300 element set are there with exactly 5 members?

Snk(300,5) = 4090911221081438794246476628982333220442638220478190976605825883041810075846210707549539683575840119419320444839733557924380958558669263052623178486192842007420476226080784639399165079562132928175942227395000 set partitions.

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Attendance Question 5:

Is [1,1,4,11,41,162,715,3425,17722,98253,580317,3633280,24011157] in the OEIS? What is the A-number?

Yes, it is A296.

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Attendance Question 6:

Is [1,1,4,10,40,140,630,2800,14070,...] in the OEIS? A number?

This sequence does not appear to be in the OEIS.

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Attendance Question 7:

Let a_k(n): Be the number of permutations where every cycle is of length at least k.

a_1(n) = n!, a_2(n) = d(n)

For k=3,4,5, ... find the first 30 terms of a_k(n) and see whether they are already in the OEIS? Which k is the smallest that is NOT there? Should it be?

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k = 3:

No cycles of length 2 or less: x^3/3 + x^4/4 + ... = -log(1-x) - x - x^2/2.

egf := 1/(1-x) * exp(-x) * exp(-x^2/2);
[seq(coeff(taylor(egf,x=0,31), x, i) * i!, i = 1..30)];
[0, 0, 2, 6, 24, 160, 1140, 8988, 80864, 809856, 8907480, 106877320, 1389428832, 19452141696, 291781655984, 4668504894480, 79364592318720, 1428562679845888, 27142690734936864, 542853814536802656, 11399930109077490560, 250798462399300784640, 5768364635100620089152, 138440751242507472273856, 3461018781064593367693824, 89986488307675206245836800, 2429635184307185219369763200, 68029785160601345467104670848, 1972863769657440129000783404544, 59185913089723198139150966450176]
It is in the OEIS as A38205.


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k = 4:

No cycles of length 3 or less: x^4/4 + x^5/5 + ... = -log(1-x) - x - x^2/2 - x^3/3.

egf := 1/(1-x) * exp(-x) * exp(-x^2/2) * exp(-x^3/3);
[seq(coeff(taylor(egf,x=0,31), x, i) * i!, i = 1..30)];
[0, 0, 0, 6, 24, 120, 720, 6300, 58464, 586656, 6384960, 76471560, 994831200, 13939507296, 209097854784, 3345235180560, 56866395720960, 1023601917024000, 19448577603454464, 388972171805410656, 8168409582839579520, 179704944537482689920, 4133213636880538425600, 99197131945856677419456, 2479928332674564111757824, 64478136551481363914841600, 1740909683073851861225241600, 48745471108643784941792976000, 1413618662349077210550915116544, 42408559873877916528445525618176]
It is in the OEIS as A47865.

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k = 5:

No cycles of length 4 or less: -log(1-x) - x - x^2/2 - x^3/3 - x^4/4.

egf := 1/(1-x) * exp(-x) * exp(-x^2/2) * exp(-x^3/3) * exp(-x^4/4);
[seq(coeff(taylor(egf,x=0,31), x, i) * i!, i = 1..30)];
[0, 0, 0, 0, 24, 120, 720, 5040, 40320, 435456, 4959360, 60255360, 782939520, 10870104960, 162642864384, 2602111599360, 44265739714560, 797239080529920, 15148603957800960, 302953165014675456, 6361636837067089920, 139952704431008286720, 3218922790460015063040, 77254605493692206469120, 1931370859803315548749824, 50215664438816457218457600, 1355822305759073660955033600, 37963012534836747190338355200, 1100927264115029271588399513600, 33027818402324374018424155570176]
It is not in the OEIS so k = 5 is the smallest k such that the sequence is not in the OEIS.

I think it could belong in the OEIS, ensuring there are no cycles of length 4 or less is still a relatively small restriction so there could potentially be some use with this sequence.