#ATTENDANCE QUIZ FOR LECTURE 19 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p19
#with an attachment called
#p19FirstLast.txt
#(e.g. p19DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. 13, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 8

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

1) 
What is the explicit expression for the sequence
a(0)=0, a(1)=0, a(2)=0, a(n)=1 for n = 3,4, ...
f(x)=0+0+0+x^3+x^4+x^5+...

Use Maple to find an explicit expression for the generating function of 
0,1,8,27,64,125,...,a(n)=n^3

sum(x^n*n^3, n = 0 .. infinity)
	x*(x^2 + 4*x + 1)/(x - 1)^4

2) Find the EGF of a(n)=n, for 0<=n<=5, a(n)=0 if n>=6
0,1,2,3,4,5,0,0,0,...

0*x/0!+1*x/1+2*x^2/2!+3*x^3/3!+4*x^4/4!+5*x^5/5!
=0+x+x^2+x^3/2+x^4/6+x^5/24

3) What is the EGF of a(n)=0 for n=0,1,2,3,4,5 and a(n)=1 for n>=6
0,0,0,0,0,0,1,1,1,...
0+0+0+0+0+0+x^6/6!+x^7/7!+x^8/8!+x^9/9!+...
=exp(x) - x^5/120 - x^4/24 - x^3/6 - x^2/2 - x - 1

4) Find the EGF of a(0)=0, a(1)=0, a(n)=(n-2)! for n>=2
0,0,0,1,2,6,24,120,...
0+0+0+x^3/3!+(4-2)!x^4/4!+(5-2)!x^5/5!+...
=(x - 2)*exp(x) + x + 2

5) What is the A-number of this sequence?
1,1,4,38,728,26704,1866256,251548592...
A001187

How many digits does the number of labeled connected graphs
with 150 vertices have?

10237679866613227689933966670078313443150043132853076906286464580162975316853677173177310590862106268927966079623397135234574364986806971998505744804516420114879839684936331772044141241976248133100789617901466906076056647995584686898165579874224400195550418621657599887307182534736311702295520309280470173143063202535320197918793788123330881271595669984163071912764545723178727014206344921648094040027007522283290304275430056814954257311428281943549877772625609033175388791853784210081841124194091541870448704782938393756397594653304611738777465484934381234280952644376525251074218052135732602315906061254822922271572666050503251028778145484484755315029010674746858581260308572883178732304310594971118010798026163487552029999617180796489870244320155510769701611740565079958379119412966382133456516914858242363228378390818237171956637083229694379044094480925855977341670365164213336480396037969636545560624089084106037496142833180644995003162240308231866989382501493268553727195439019455786059125396803763911243779855810193912079980624038695049208084423448666031074475674807500912167373005323505098904801547171003826341367718785623389091506537103867943363286340114987643897514769125023287884202433855251691157699923308389098465192600683870630834809704035916870388438670494975962035497659625665302414807757926839604695223572605457485614230659978415560657083390158597003997208582715993492242549604467740802474894702139653404608268440252359965240127059007360470430855247189277567961911781784049643155814713723181638565783601072673063249144997079607857214479044272804835090983000980975648593551206604842501500300561446603826872948397524665768676620734324612612675474050235655104681791728627284373970247074678137520773441875528829191673350771155488779722931923818702892888319036846528351750755609424272203616876109880892728337640387656520027223240300137553862874883269229663344212365660860368178505547662596322721124410472994137418943071095546421687881650228663934915594944674545297595108470509635936775746695932818118586614440253382376753717596313359491483287361663634954807095663522147513709616643079909795861272226456836110837446213399718722430072732104689492365140372324319151340725327945869821158369924604115285625846867762978467428006477955582399308817055316885152684357947636609333134641386750327257545466265382164227692785441344239111564670264111827545462251281744924527260522196239759176067961184518755045454608706028534825951365921844039873294267182113786788746144907789630501084920198998565698951187789691742820326926325550477277857109543415874790014209016240199533322651364322016859175636900522059802749274497961342284305845778042062744567510939080821648070823173256068928697760095895147556458848578804851436660231899363687460749659879754672407462488771737755152240670787437034397415631172686578683430772068014520298303034186169063774697050978747256364410290174115496024459993606682184632144244238658473467760942037480985220165895565924570066672494255957020684664801447616243643169747241177451401559393053492020378661381495999755470905801824948565737390560871350732829784978101763257536871142822444833681032414299805024754408512460500035195493404280541184999668777276994027282683231187539152169794835486485949015680561179515381627213593155657819615310469950515760637846424603130947498376603134307235731663726360464060070511728378754705040534724490510397865984

6) Use the technique of weight-enumeration to find the exact number
of sequences a[1], a[2], ..., a[r] (r can be any length)
where each of the a[i] is a member of {3,4,7} that add up to 1001

|A|_x = 1 + x^3|A|_x + x^4|A|_x+x^7|A|_x
f(x)=1/(1-x^7-x^4-x^3)
37327228629056835260060479719213438237141848693395064396004356099234232222305300761357555310103620202925186169

7) 
Is [{{1,3,4},{6,7}}, 52] a member of X(7)?
Why?

Yes, labels are distinct and length(pi)+size of set being partitioned = 7. However, the order of the permutation and the setpartition in the data structure is reversed from the order used in the lecture example; I'm not sure if that was intended.

8)
Later on we will prove that the egf of Labelled trees is 
Sum(n^(n-2)*x^n/n!,n=1..infinity)

How many triples of the form 
[labeled tree, permutation, setpartition]
of size 150 (meaning the number of vertices of the tree + the length of
the permutation + size of the set that setpartition partitions is 150)

egf(X)=egf(LABELED TREE)*egf(PERMUTATION)*egf(SETPARTITIONS)

egf(permutations) = Sum(n!*x^n/n!,n=0..infinity)=1/(1-x)

egf(setpartitions) = exp(exp(x)-1)

I looked up the number of labeled trees for size n and found that it is n^(n-1). I will use this number, hopefully it's right.
egf(labeledtree) = sum(n^(n-2)*x^n/n!,n=0..infinity

egf(f)=1/(1-x)*exp(exp(x)-1)*sum(n^(n-2)*x^n/n!,n=1..infinity)

coeff(taylor(1/(1-x)*exp(exp(x)-1)*sum(n^(n-2)*x^n/n!,n=1..infinity),x=0,151),x,150)*150!;

28977095542425090391080657989766683580624301002611004688373123464683818218205152245237725058499348410444136844355627930505248173365633049546567636265686241354967124715400767418627896833174732525183101768243433321734433805222261424610837985663984077225116660868853206011638934836218209026388216963202293429209093172742679375