#OK to post homework
#Karnaa Mistry, 11/22/20, Assignment HW #21

with(combinat):

# 1.

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




# 2.

# I tried doing 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)
# I used this because I wanted to combine the weight enumerator expression for edges a^m with the taylor expansion in 
# the procedure NuKcomp, which involved having the ^k/k! to find the number of components.  However, when I evaluated
# this, I got 0 all four times, so I am unsure as to what might have went wrong in my combination of the two
# procedures' functions.

# Alternatively, using my proc WtEdConGClever2 in part 6), which uses the given weight-enumerator in Question 6, I was able to nonzero answers like
# coeff(WtEdConGclever2(50,k,a)[50],a,70), such as k=2 ==> 2267762282072205271778791270680596849921516338574571823883456519281730035101965414722739023408444177334403072000000
# So I would need to find this weight enumerator using taylor methods if I was to derive it manually without the given in #6




# 3.

# FindCutOff(30,1.05) gives 84.  And my EsimateCutOff(30,j) for 'large' j (1000 was taking very long) also ends up in the 80s, ex. EstimateCutOff(30,100) gave
# values within +4/-4 of 84 (like 80,83) on more than one run




# 4.

# seq(evalf(FindCutOff(10*i,1.05)/(10*i)),i=2..6) = 2.450000000, 2.800000000, 3., 3.160000000, 3.300000000
# I couldn't find a specific pattern from this sequence, I first thought that maybe it has to do with the change from +0.35, to +0.2, +0.16, +0.15,
# but this doesn't resemble a specific pattern, either.




# 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