#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

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



Question 1:


When was Rabbi Levi Ben Gerson born? When did he die?


Answer 1:


Born: 1288, France
Died: 1344


Question 2

When was E.T. Bell born? When did he die? What is he famous for? What is the name of his famous book? He was also a writer, what was his pen name?


Answer 2:

Born: Feb 7th 1883
Died: December 21, 1960

Pen Name: John Taine under which he published fiction

Famous Book: Men of Mathematics


Question 3:

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


Answer 3:

SnkSeqC(301,5)[300]
4090911221081438794246476628982333220442638220478190976605825883041810075846210707549539683575840119419320444839733557924380958558669263052623178486192842007420476226080784639399165079562132928175942227395000




Question 4:

Is this sequence in the OEIS? 
A number?
Description?
[0,1,1,4,11,41,162,715,3425,17722]


Answer 4:

A000296

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




Question 5:

Is this sequence in the OEIS? A number? 

[0,1,1,4,10,40,140,630,2800]


Answer 5:

This sequence is not in the OEIS. 



Question 6:

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 it is in the OEIS. What is the smallest value of k for which the sequence is not in the OEIS? Should it be in the OEIS?

 

Answer 6:


For k=3: no cycles of length 2

x^3/3 + x^4/4 + x^5/5 + x^6/6+ .... = -log(1-x)-x-(x^2/2)

f:= exp(-log(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, 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



For k=4: no cycles of length 3

f:= exp(-log(1-x)-x-(x^2/2)-(x^3/3))

f := exp(-ln(1-x)-x-(1/2)*x^2-(1/3)*x^3)
f:=taylor(f,x=0,31):
[seq(coeff(f,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]

A047865



For k=5: no cycles of length 4

f:= exp(-log(1-x)-x-(x^2/2)-(x^3/3)-(x^4/4))

f := exp(-ln(1-x)-x-(1/2)*x^2-(1/3)*x^3-(1/4)*x^4)
f:=taylor(f,x=0,31):
[seq(coeff(f,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]


Not in the OEIS!!

The smallest k for which the sequence is not in the OEIS is k=4!