#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: 6

#1. when was Rabbi Levi Ben Gerson born and died?

#He was born at Bagnols in Languedoc, France at 1288 and died at 1344.

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#2.when was E.T. Bell born and died? what was he famous for? and his famous book?

#Eric Temple Bell was born in Scotland in 1883 and died in 1960. His work is famous in number theories including
#Bell series. And he published famous book named "Men of Mathematics"

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#3. In addition to being a brilliant mathematician, Bell was good writer. what was his pen name?

#He used the pen name John Taine for his fictional books


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#4. How many set partition of a 300-element set are there with exactly 5 members.

#basically a sterling number of second kind
#Snk(300,5)
#4090911221081438794246476628982333220442638220478190976605825883041810075846210707549539683575840119419320444839733557924380958558669263052623178486192842007420476226080784639399165079562132928175942227395000

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#5, Is this set [1, 1, 4, 11, 41, 162, 715, 3425, 17722, 98253, 580317, 3633280, 24011157,...] in the oeis?
#What is this A-number?

#This is called Set partitions without singletons i.e. number of partitions of an n-set into blocks of size > 1
#and its A-number is A000296


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#6 is this set [0,1,1,4,10,40,140...] in the oeis?

#no. Its not in the oeis database

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#6. let a_k(n): Be the number of permutation where every cycle is of length at least k
#a_1(n)=n!, a_2(n)=d(n)
#for k=3,k=4,k=5 find the first 30 terms of a_k(n) and see whether there are 
#already in the OEIS? which k is the smallest that is Not there?
#should it be?

#answer: 
#for k =3 , egf should be exp(-(log(1-x) -x-x^2/2) = 1/(1-x)*exp(-x)*exp(x^2/2)
f3:=exp(-x)*exp(x^2/2)/(1-x);
f:=taylor(f3,x=0,31):
a_3(n) = [seq(coeff(f,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]
#A038205, Number of derangements of n where minimal cycle size is at least 3.

#for k=4,5 we use same tactics
#k=4
f4:=exp(-x)*exp(-x^2/2)*exp(-x^3/3)/(1-x);
f:=taylor(f4,x=0,31):
a_4(n) = [seq(coeff(f,x,i)*i!,i=1..30)];
a_4(n) = [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]
#A047865,Number of derangements of n where minimal cycle size is 4.

#k=5,
f5:=exp(-x)*exp(-x^2/2)*exp(-x^3/3)*exp(-x^4/4)/(1-x);
f:=taylor(f5,x=0,31):
a_5(n) = [seq(coeff(f,x,i)*i!,i=1..30)];
a_5(n) = [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]
#this is not in the OEIS