#OK to post homework
#Joseph Koutsoutis, 02-02-2025, Assignment 3

read `C3.txt`:

#4

PC:=proc(n,p,K) local i:evalf(coeff(add(IsCo(RG(n,p)),i=1..K),true,1)/K):end: 


# TestPC prints the output of PC(n,p,200) for n=10,20,30 and p=1/10,2/10,...
# In C3.txt, I changed the error checking of LC from:
#       if not (type(p,fraction) and p>=0 and p<=1) then
# to:
#       if not (type(p,rational) and p>=0 and p<=1) then
# since this was returning fail when I inputted p=10/10.
TestPC := proc() local n,p,K:
  K := 200:
  for n in [10,20,30] do:
    for p from 1 to 10 do:
      printf("n = %d, p = %d/10, K = 200, PC(n,p,K) = %f\n", n, p, PC(n,p/10,K)):
    od:
  od:
end:

# Here is the output of TestPC:

# n = 10, p = 1/10, K = 200, PC(n,p,K) = 0.005000
# n = 10, p = 2/10, K = 200, PC(n,p,K) = 0.230000
# n = 10, p = 3/10, K = 200, PC(n,p,K) = 0.640000
# n = 10, p = 4/10, K = 200, PC(n,p,K) = 0.945000
# n = 10, p = 5/10, K = 200, PC(n,p,K) = 0.985000
# n = 10, p = 6/10, K = 200, PC(n,p,K) = 1.000000
# n = 10, p = 7/10, K = 200, PC(n,p,K) = 1.000000
# n = 10, p = 8/10, K = 200, PC(n,p,K) = 1.000000
# n = 10, p = 9/10, K = 200, PC(n,p,K) = 1.000000
# n = 10, p = 10/10, K = 200, PC(n,p,K) = 1.000000
# n = 20, p = 1/10, K = 200, PC(n,p,K) = 0.035000
# n = 20, p = 2/10, K = 200, PC(n,p,K) = 0.735000
# n = 20, p = 3/10, K = 200, PC(n,p,K) = 0.975000
# n = 20, p = 4/10, K = 200, PC(n,p,K) = 0.995000
# n = 20, p = 5/10, K = 200, PC(n,p,K) = 1.000000
# n = 20, p = 6/10, K = 200, PC(n,p,K) = 1.000000
# n = 20, p = 7/10, K = 200, PC(n,p,K) = 1.000000
# n = 20, p = 8/10, K = 200, PC(n,p,K) = 1.000000
# n = 20, p = 9/10, K = 200, PC(n,p,K) = 1.000000
# n = 20, p = 10/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 1/10, K = 200, PC(n,p,K) = 0.285000
# n = 30, p = 2/10, K = 200, PC(n,p,K) = 0.950000
# n = 30, p = 3/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 4/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 5/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 6/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 7/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 8/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 9/10, K = 200, PC(n,p,K) = 1.000000
# n = 30, p = 10/10, K = 200, PC(n,p,K) = 1.000000

# According to Wikipedia (https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model#Properties_of_G(n,_p)),
# ln(n) / n is the threshold for connectivity, which agrees with the results above.

#5

NuCGx := proc(n,x) local f,z,i:
option remember:
f:=log(add((1 + x)^binomial(i,2)*z^i/i!,i=0..n)):
f:=taylor(f,z=0,n+1):
[seq(i!*coeff(f,z,i),i=1..n)]:
end:


#6

NuCGk := proc(n,k) local x, polys, i:
  polys := NuCGx(n, x):
  [seq(coeff(polys[i], x, i-1+k), i=1..n)]:
end:

# Here are some values of k and their corresponding OEIS numbers:
# k =  0, OEIS number A000272 (https://oeis.org/A000272)
# k =  1, OEIS number A057500 (https://oeis.org/A057500)
# k =  2, OEIS number A061540 (https://oeis.org/A061540)
# k =  3, OEIS number A061541 (https://oeis.org/A061541)
# k =  4, OEIS number A061542 (https://oeis.org/A061542)
# k =  5, OEIS number A061543 (https://oeis.org/A061543)
# k =  6, OEIS number A096117 (https://oeis.org/A096117)
# k =  7, OEIS number A061544 (https://oeis.org/A061544)
# k =  8, OEIS number A096150 (https://oeis.org/A096150)
# k =  9, OEIS number A096224 (https://oeis.org/A096224)
# k = 10, OEIS number A182294 (https://oeis.org/A182294)
# k = 11, OEIS number A182295 (https://oeis.org/A182295)
# k = 12, OEIS number A182371 (https://oeis.org/A182371)
# k = 13, no OEIS entry