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

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#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




Question 1:

(i) What is an explicit expression for the sequence a(0)=0, a(1)=0, a(n)=1 for n=3,4...


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


Answer 1:

(i) 

assume(abs(x)<1)
sum(x^n,n=3..infinity)

-x^3/(x-1)

(ii) 

f:=sum((n^3)*(x^n),n=0..infinity)

f := x*(x^2+4*x+1)/(x-1)^4




Question 2:

Find EGF of a(n)=n for 0<=n<=5, and  a(n)=0 if n>=6


Answer 2:

EGF := (0*x^0/0!) + (1*x^1/1!) + (2*x^2/2!) + (3*x^3/3!) + (4*x^4/4!) + (5*x^5/5!)



Question 3:

What is the EGF of a(n)=0 for n=0,1,2,3,4,5 and a(n)=1 for n>=6 ?


Answer 3:

EGF := (x^6/6!) + (x^7/7!) + (x^8/8!) + ...

And finding the explicit form using maple, 

f:=sum((x^n)/(n!),n=6..infinity)
f := exp(x)-(1/120)*x^5-(1/24)*x^4-(1/6)*x^3-(1/2)*x^2-x-1




Question 4:

Find the EGF of a(0)=0, a(1)=0, a(n)=(n-2)! for n>=2


Answer 4:

Using maple, 

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

f := -x*ln(-x+1)+ln(-x+1)+x



Question 5:

(i) What is the A number of [1,1,4,38,728,26704,1866256,251548592,...]

(ii) How many digits does the number of labeled connected graph with 150 vertices have?

Answer 5:

(i) A001187

(ii) 

coeff(taylor(log(add(2^(n*(n-1)/2)/n!*((x^n)/n!),n=0..155)),x=0,155),x,150)*150!


114768360602673712250519063637898847290958746538929685056259811627281123361821119718741539615905154875739973585042994246558584590753903713943544157485700194673679851736886998487801045374926163193390134873860553391021687513552049699577449830538626542721979171227712499428546990625876623533200183515734762671592737303438415646148785170713291833748893550442131288510022091477691137101919622104014031224191136044062902244323898760073601062038895560698117175117353408111338919319654774325484012232525015919309307648472298176248929053651223977870007619381000025441943846804780140825112063055811547854402228831382806241616381194251737210932836362460980935831391202040783540272852179719275289530030395328583199854546333686351930675373612856347756663597199631375412312853811430127678588278338351114101013065547071252001173636833478176017040289465122781537396939307926792896488545165739023261095708941010093859198458444313493733304908020113830049932660784847712288449692626115648854537442434365189802048123043056904449271972571051878811031199667282791076505138728159940430960646823984570651456135852798434854912225299796554641250070364106464358870886082579413460641083212237462012278131436393104014669359293430254995302452137272247030529882534932029143064134031375346389956784371074589870739920244618750624098068772607472651087843395797166156604066001866540762718819614118331955739026960732296147297992698581038129659100890618637508076634746928868642905772679069756614379945042460085438749315928661824376080753490128683506476497591589078753210020712791229804941702078705680339704154705622600863979117287352064490690061018529881311357224414803454888829442246600180991039376177494272662720727801926323076991989979482824718718041202115441692712076896513980409547785195508907837763550951461397602745422556677838239261185854923216033397564200696173791377500171561096080643104088998890380484284757155888264020976444602821036874131423050902598365806426892168801115527095356580669142485057344911221250609456242434456168113420534647503279352471489941513595900644756529409608988337191405442047257888011893014381193364697318154167103206442203083299145592671338241440641645799866512281971874356074165674997361395391851370842813486814966437171861207570386858891735324775112822525664220614921311481007803416951459413230811117025073621153307666696656498586041526221580627171142539915417257178417774606228949016674263178543095369147607672596785120530138721644504797835946476162858996896060029360562616510163050016971099988059328694247290901232424720519405506815852104218508630491158543349300560351921307910356563861741169513774001895164830746022880907023179188957845045332962701271804623046528731378835212668678555516427968947382705809846160037334132971195076174872229760273638400282053849791854402318258662053277886388678536039622544425637761549954195902598940835409182269752410240701724873862246486899019777676806913750660996703477490127027433502205951045857742839602283580067144700510251650442773208036942909579874219413995198140706286390023739875255349706238974681281580310798602035415667047927898587859530808710994246212704779040901710665203651684575790281680356807549364816051747192668088935286554373091389291718366987075592613360176852203110372527414712874482640376455254883696423906218960665913396984863840358269270018578357739857679220736/640492493141249886288067962515029791985968964724779138980122433145476734300073667407785444774143814079719104241654363428140338389713347010741460456581357322097094221771420592282552975514980456573539413511753082275390625






Question 6:

Use the technique of weight enumeration to find the exact number of sentences a[1], a[2],...a[r] (r of any length) where each of a[i] is a member of {3,4,7} that's adds up to 1001.


Answer 6:

Let A={3,4,7}

Then 

|A|_x = 1 + x^3*|A|_x + x^4*|A|_x + x^7*|A|_x

|A|_x = 1 + (|A|_x * (x^3 + x^4 + x^7))

|A|_x * (1- (x^3+ x^4 + x^7)) = 1

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

Finding the Taylor expansion and the coefficient of x^1001,

coeff(taylor(f,x=0,1005),x,1001) * 1001!

15035021255571750668745147425143355983803589356057923279229565679137921082255841996771689237653322993397835660310845468402249753230859080705945977317804127508498156457758569303081548801855293294933012457779384340096039344890915059276242743115366267624577991168219817531470273292475794918139026342956369342751702654338978647031568440614967208505159420165840623649020337033914396527251359878599136961535769555544582886460905511147061541748694698250691136378148407203872581176757287105351239823200421640111958680943681937689472359731834504120416743891386060932768537622709901784830810114124323029817066801978113973038473770994856712073371587764546054785170717336668479884662042622823870010190583977266804170043417899459663792347865709297199396426213501885769981783716745312053446493979753682932092333268833577770116220546983062729357324016342994531254966643131114850035115650639094240130625537545378120952451020482037786797770387090016379900044200676816370608822963379905373710556833084473082941889873257778186497951013458173740151597559977050325126755018024607921148175772163292534842639318057700013750353708344208534517733177761857070487085302637211514075562128795083107454395920973755742999400829371801886849762008635356109256150451666080674619100528755971005640419712266440663246922035998850065777334152395258039567549077368390549214557927258875502369947397956743545628150530284923107922677048167444772284697491926404449311240011689143474039618799577053036279534042846586633395897301659523375908610164941451492459520767016094622874823132951949011104653399865581665974525701690492878085218669229338286721676785278455931578742289884594336596344475762560852742067895009740533429445223899060797643221390988656479226539068288650474693261505536025251549325416314158425322023704953930589777190100792987856917244244903859528113498427971826901026696973509383431120238410896205243826756168152554771406321987494367052254193960865932759266338198445640719909376951589068216808251738366228816738061636235097641871596198589250697604704861826857376734139619670134929869556775062996733272789537807161524007250011976511122177063598450847619612404576204808119449600103731042568785052997998121329725932155658816572715430215998725167775251191722053725046207113829258280194492529080271852840455855359899203524140807066747511917641390079427104068054825803123603702080318683040513976294154675205379682875620004518064450286380209803156281243794838564896768000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000




Question 7:

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

Answer 7:

(i) No. Because we define an element of X(n) to be of the form [permutation pi, SetPartition SP] not 
[SetPartition SP, permutation pi]



Question 8:

How many triplets of the form (tabled tree, permutation, set partition) of size 150 (meaning that the number of vertices of tree + elements of permutation + size of set partition is 150)


Answer 8:

Multiplying the EGF's of tabled tree, permutation, set partition, we get

coeff(taylor((1+sum((n^(n-2)*x^n)/n!,n=1..infinity))*(exp(exp(x)-1)/(1-x)),x=0,152),x,150) * 150!


28977095542425090391080657989766683580624301002611004688373441982498475144258790949730800839265420513593407126434947114816932428043259928872950627837484054712104869467896270548805752624407025846130055117327436957879166244375004554236854437226696641804705690475416777108324895809564401177362712104654213569067621944677124028