#OK to post
#Soham Palande, Assignment 20, 11/22/2020



#PART 1

(i) There are exactly 7 components, and each component has odd size

f:=add(x^(2*i+1)/(2*i+1)!,i=0..52)^7/7!:

coeff(taylor(f,x=0,101),x,100)*100!

0

(ii)  There are exactly an odd number of components, but the sizes of the components can be any strictly positive integer


f:=exp(x)-1:

coeff(taylor(add(f^(2*i+1)/(2*i+1)!,i=0..50),x=0,101),x,100)*100!;

23792695638183628316167125166708722722054741318989995555688894753342022292474103758734144789060946967960716668504295


(iii) There are exactly an odd number of components, but the sizes of the components can be any strictly positive integer except that we do not allow components of size 2 and 5

f:=exp(x)-1-((x^2)/2!)-((x^5)/5!):

coeff(taylor(add(f^(2*i+1)/(2*i+1)!,i=0..50),x=0,101),x,100)*100!;

537911979212289319210315022167976910293816540852093547715022508541682212897278245024355264789522538287080267265






#PART 2

How many permutations are there of {1,2,..., 100} such that


(i) There are exactly 7 cycles, and each cycle has odd size


f:=add(x^(2*i+1)/(2*i+1),i=0..52)^7/7!:

coeff(taylor(f,x=0,101),x,100)*100!

0


(ii) There are exactly an odd number of cycles, but the length of the cycles can be any strictly positive integer


f:=-log(1-x):

coeff(taylor(add(f^(2*i+1)/(2*i+1)!,i=0..50),x=0,101),x,100)*100!;

46663107721972076340849619428133350245357984132190810734296481947608799996614957804470731988078259143126848960413611879125592605458432000000000000000000000000


(iii) There are exactly an odd number of cycles, but the lengths of the cycles can be any strictly positive integer except that we do not allow cycles of length 2 and 5

f:=-log(1-x)-((x^2)/2)-((x^5)/5):

coeff(taylor(add(f^(2*i+1)/(2*i+1)!,i=0..50),x=0,101),x,100)*100!;

23172213523966770724123670280303621010635693971914974901987101498394969134506826670931406322436188692149594499306436597877538608022152025529531964918909222550




#PART 3

(i)   How many labeled connected graphs are there with 100 vertices? 
(ii)  How many are there with exactly 2 components? 
(iii) With exactly 100 components?


(i)

f:=log(add(2^(n*(n-1)/2)*x^n/n!,n=0..101)):

coeff(taylor(f,x=0,101),x,100)*100!

125452273648412141401139200773089339811437122262323352216462286700734387627331522885709312904647210371328794107858418293356140610389233010890575674923332327423429917033308833625087865757067009121643735335562957720919464360279849562024268481514680519466433342816295305461792608710076067024418040646132922593872156786882042514851134725907407734998659047116141535192232282791034794601497425933135733739224548153792567839779141913680549722028773475603674523853169051845195421079685934332310882053907313995168923535704556709136009686053259752740308577712484995569783650846269720521608249582420017860918845622644663874987575160515268766971368721963159046529551370104575096202684311358624098672672476994761800126366704378281769378475834662925454186722285874859694872242026068795039642380038691501013387521486053366727435218916045301026547623117868196574350026663672938544838421549439678291079144887169351257353939537143880911795509769485297854345823362270660712007071953958048732807833151699233932553517365602355625760928908188344900652899469143373311896574182769657272933821876553379518700197938172155010185719146936187180903250543579433005278984692348282568379624191843385007057047580846112172322370992146748764340302424100769640802926078370838482052406435490036434092467614465282934547622486501083341846045800849550829377015688082924233744178977266360888077466658752803757886543953371541948150838050857811138739570190403175194723845806878286855221273591078542964296886759202533689916185534005248

(ii)

coeff(taylor(f^2/2!,x=0,101),x,100)*100!

19792878830464097434146485069739463865673187370672887935834214122103592997909477467639145015413302778571558698199938989875722363632867076525500775193313203219625694859620492756307560817786124480904571667305985231399040766859432538780170724152051889531424530851027515639435541861971440089962657458762079240342770560135993633520904674629099681471079341302098763093130534058092714306341148454323169655736417188716167485638876856164406937663195315675056473320830072088996900023615988558834190601024551606490996047256677952803727152702099973371018028710701366236803289614033811563475696670364580819031243080945830466884602280114938472440443402744527681562033913068944126647000531086090506590597863434313244427618995892382119045374120085198882365703112462192132262796290715887651629501613500765319382697459714625232601391113377456170319146465206446792331217136689657731214890373782128820154293953635221308380687242012269185806693761170244502543846560216890121849276503785641853915478872687428718708183133628359872764435735322526121955322100244200579269171473679314961687680638065276652910066401714258899512328773662068014373263170965092624777284239415572512416129024249860116530210873861235283964287841875225918264476127609182676153172047392096718128978027994944226801334642136592881198292406724597930004937454732160477796576767954616879358825917759244042876084345101045377871469937328841023216205534951719728974096997883111102605252071058831339598818474925540591534080


(iii)

coeff(taylor(f^100/100!,x=0,101),x,100)*100!


1