#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: 7

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

1) When was Rabbi Levi Ben Gerson born? And when did he die?

He was born in 1288 and died in 1344.

2) When was E.T. Bell born and when did he die? What is he famous for? He wrote a classic book on the history of mathematics. What is the name of the book that E.T. Bell wrote?

He was born February 7, 1883 and died December 21, 1960 and famous for his Bell numbers/Bell polynomials. His most famous book is probably Men of Mathematics. 

3) In addition to being a brilliant mathematician he was also a writer and he had a pen name. What was the pen name of E.T. Bell?

His pen name was John Taine.

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

egf of atoms x^5/5!

f:=taylor(exp(x^5/5!),x=0,301):
coeff(f,x,300)*300!

652759846354937088556719084676441820841461277955440992947509873566186746926913477804190102993738747288558962817342280333376601229437981981830595329045912585294619756668990529039361421955723871166709352362795052056089329148797166961882653114077768994187647185241827325815215774834594663417856252836646111829015715654359914527131889123171510853566878325686289541262450695986313848237131721332315997927035109376

5) Is this sequence in the OEIS? What is the A number?
0,1,1,4,11,41,162,715,3425,17722,98253,580317,3633280,24011157...

Yes, A000296.

6) Is it in the oeis? A number?
0,1,1,4,10,40,140,630,2800,14070,73150,412720,2422420,15095080...

It is not in the OEIS.

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,k=4,k=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?

k=3
egf of atoms: 
f=x^3/3+x^4/4+x^5/5+...=-ln(1 - x) - x - x^2/2

f:= -ln(1 - x) - x - x^2/2;
f:=taylor(f,x=0,31):
seq(coeff(f,x,i)*i!,i=1..30)];


[0, 0, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000, 620448401733239439360000, 15511210043330985984000000, 403291461126605635584000000, 10888869450418352160768000000, 304888344611713860501504000000, 8841761993739701954543616000000]

It is there, A124355.

k=4 
egf of atoms
x^4/4+x^5/5+...=-ln(1 - x) - x - x^2/2 - x^3/3

f:= -ln(1 - x) - x - x^2/2 - x^3/3;
f:=taylor(f,x=0,31):
seq(coeff(f,x,i)*i!,i=1..30)];

[0, 0, 0, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000, 620448401733239439360000, 15511210043330985984000000, 403291461126605635584000000, 10888869450418352160768000000, 304888344611713860501504000000, 8841761993739701954543616000000]

k=4 is not there. Should it be? I think if it is something important and useful that comes up often enough it should be included.