#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. p20DoronZeilberger.txt)

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


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 6

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




#ATTENDANCE Q. #1 for LECTURE 20
#When was Rabbi Levi Ben Gerson/Gersonides born? When did he die?

#ANSWER to Q. #1:	

# He was born in 1288, and he died on April 20 1344



#ATTENDANCE Q. #2 for LECTURE 20
#When was E T Bell born? When did he die? What is the name of that famous book?
#He was also a writer, what was his pen name for his fiction?

#ANSWER to Q. #2:

# He was born  February 7, 1883 and died December 21, 1960
# A famous book he had on the history of mathematics is called 
# "Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincaré"
# His pen name was John Taine




#ATTENDANCE Q. #3 for LECTURE 20
#How many set partitions of a 300 element set are there with exactly 5 members?

#ANSWER to Q. #3:

#Snk(300,5) = 
#40909112210814387942464766289823332204426382204781909766058258830418100758462107075495396835758401194193204448397335579243809585586692#63052623178486192842007420476226080784639399165079562132928175942227395000





#ATTENDANCE Q. #4 for LECTURE 20
#Is that sequence 1,1,4,11,41,162,... in the OEIS?

#ANSWER to Q. #4:

#Yes, A000296: "Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. 
# Also number of cyclically spaced (or feasible) partitions."




#ATTENDANCE Q. #5 for LECTURE 20
#is the sequence 0, 1, 1, 4, 10, 40, 140, 630, ... in the OEIS?

#ANSWER to Q. #5:

It does not seem to be in the OEIS, there can be many, many other ones like this, too, so maybe
the OEIS does not need so many. (choose some atomic sizes)




#ATTENDANCE Q. #6 for LECTURE 20
# 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 if they are in the OEIS
# Which k is the smallest that it is not there? Should it be?

#ANSWER to Q. #6: (I only went up to 15 for spatial reasons)

(for each one we change -ln(1-x)-x by subtracting x^2/2, then x^3/3 and so on)

#a_3(n): 0, 0, 2, 6, 24, 160, 1140, 8988, 80864, 809856, 8907480, 106877320, 1389428832, 19452141696, 291781655984
#This is in the OEIS, A038205

#a_4(n): 0, 0, 0, 6, 24, 120, 720, 6300, 58464, 586656, 6384960, 76471560, 994831200, 13939507296, 209097854784
#This is in the OEIS, A047865

#a_5(n): 0, 0, 0, 0, 24, 120, 720, 5040, 40320, 435456, 4959360, 60255360, 782939520, 10870104960, 162642864384
#This does not seem to be in the OEIS.  It might be interesting to have it in, since some of i=4..8 it is close to
#the factorial numbers, so a partial search may bring up multiple results of different kinds