#OK to post homework
#Hari Amoor, November 22, 2020, Homework Assn. #21

# Question 1

# coeff(WtEdConGclever(50,a)[50],a,70) = 557570125262936299552095051452832858951958306952392379552719516748979534894703561052813706583449695070191616000000

# Question 2

# I am unsure of the exact solution to the problem, but I have attempted a Taylor expansion 
# of the form seq(coeff(taylor((log(add((1+a)^binomial(i,2)*x^i/i!,i=0..50+1)))^k/k!,x=0,50+1),x,70)*70!,k=2..5).
# Unfortunately, this results in a zero value, so I am not quite sure where I am going wrong.

# Question 3

# FindCutOff(30,1.05) supplies a value of 84, and my computation for EsimateCutOff(30,j) is in a similar range.

# Question 4

# seq(evalf(FindCutOff(10*i,1.05)/(10*i)),i=2..6) = 2.450000000, 2.800000000, 3., 3.160000000, 3.300000000

# I'm not seeing any pattern here; at first, it looked like logarithmic decay, but I am not quite sure at all.

# Question 5
 
EstProbKcomps:=proc(n,m,k,N) local i,co, G:

co:=0:

for i from 1 to N do

G:=RandGr(n,m):

if nops(CCs(G))=k then
 co:=co+1:
fi:

od:

evalf(co/N):
end:


# 6.

ProbKcomponents:=proc(n,m,k) local a:
evalf(coeff(WtEdConGclever2(n,k,a)[n],a,m)/binomial(n*(n-1)/2,m)):
end:

WtEdConGclever2:=proc(N,k,a) local f,i,x:
option remember:
f:=(log(Sum((1+a)^(n*(n-1)/2)*x^n/n!,n=0..infinity)))^k/k!:
f:=taylor(f,x=0,N+1):
[seq(expand(coeff(f,x,i)*i!),i=1..N)]:
end:

# ProbKcomponents(40,50,2) gave me 0.9801230118e-1 or 0.098123...
# EstProbKcomps(40,50,2,300) gave me .1066666667; this is quite close, only off by less than 1 hundredth