#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: 7

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


QUESTION #1:

When was Rabbi Levi Ben Gershon born and when did he die?

ANSWER:

He was born in 1288 and died in 1344.

QUESTION #2:

When was E.T. Bell born and when did he die? What is he famous for? He wrote a book about the history of math, what is the name?

ANSWER:

He was born on 7 February 1883 and died on 21 December 1960. He is famous for the Bell series, generating functions, Bell numbers, and Bell polynomials. The book he wrote on the history of math is called Men of Mathematics.

QUESTION #3:

What is the pen name for E.T. Bell?

ANSWER:

John Taine

QUESTION #4:

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

ANSWER:

This can be found by running Snk(300,5) which yields:
4090911221081438794246476628982333220442638220478190976605825883041810075846210707549539683575840119419320444839733557924380958558669263052623178486192842007420476226080784639399165079562132928175942227395000

QUESTION #5:

Is 1,1,4,11,41,162,715,3425,... in the OEIS? What is the A-number and meaning?

ANSWER:

Yes, its A-number is A000296. The description says: "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 #6:

Is 1,1,4,10,40,140,630,2800 in the OEIS? What is the A-number and meaning?

ANSWER:

It is not in the OEIS.

QUESTION #7:

Let a_k(n) be the number of permutations where every cycle is of length at least k. For k = 3,4,5,.., find the first 30 terms of a_k(n) and see whether they are already in the OEIS. What is the smallest k such that the sequence generated is not in the OEIS? Should it be?

ANSWER:

For k = 3, we have:
x^3/3 + x^4/4 + ... = -log(1-x) - x - x^2/2.
By our theorem, the egf is f:=(1/1-x)*e^{-x}*e^{-x^2/2}.
Then, running seq(i!*coeff(taylor(f,x=0,31),x,i),i=0..30), we get:
1, 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

Sequence for k=3 is in the OEIS with A-number A038205.

For, k=4, we have our egf, f:=(1/1-x)*e^{-x}*e^{-x^2/2}*e^{-x^3/3}.
Then, running seq(i!*coeff(taylor(f,x=0,31),x,i),i=0..30), we get:
1, 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

This sequence for k=4 is in the OEIS with A-number A047865.

For, k=5, we have our egf is, f:=(1/1-x)*e^{-x}*e^{-x^2/2}*e^{-x^3/3}*e^{-x^4/4}.
Then, running seq(i!*coeff(taylor(f,x=0,31),x,i),i=0..30), we get:
1, 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 sequence for k=5 is NOT in the OEIS. I do not feel that it should be in the OEIS because we have a full enough picture with the previous sequences. In other words, adding it to the OEIS does not give any additional information.