#OK to post homework

#Zidong Zhang, 14/2/2021, Assignment 7






#DrawMandelbrot(R,K):that plots the Mandelbrot set by picking all those complex c (of the #form x+I*y, with x,y in [-4,4] with resolution 1/R, and deciding whether to accept it if #iterating z goes to z^2+c (starting at z=0) after K iterations stay bounded. Also send the #output as .jpeg file, for R as large as possible.

###I don't know what's wrong with my code, it seems that actually generated a set of good ####point but it doesn't plot anything.


DrawMandelbrot := proc(R,K) local f,x,y,z,c,s,X,Y,S,flag,i:
S:=[]:
X :=[seq(-4+i/R, i =0..8*R)]:
Y :=[seq(-4+i/R, i =0..8*R)]:
#c:= x + y*I:
z:=0:
f:=(z,c)->evalf(evalc(z^2 + c)):
flag:=0:
for x in X do 
	for y in Y do 
		c:= x + y*I:
		for i from 1 to K do
			z:=f(z,c):
			if evalf(Re(z)^2+Im(z)^2)>4 then
				flag:= 1:
			fi:
		
		od:
		if flag=0 then
			S:=[op(S),[Im[c],Re(c)]]:
		else
			continue:
		fi:
	z:=0:
	flag :=0:
	od:
od:

plot(S):
#S:
end:

###I don't know what's wrong with my code, it seems that actually generated a set of good ####point but it doesn't plot anything.


#RandSPn(n):that inputs a positive integer n and outputs a (uniformly) at random set #partition of {1,...,n}

RandSPn := proc(n) local k:
{seq(op(RandSPnk(n,k)),k=1..n)}:

end:


#NuAdjacentMates(SP,n):that inputs a set partition, SP, of {1, ...n}, and outputs the number #of i between 1 and n such that i and i+1 are part-mates

NuAdjacentMates := proc(SP,n) local i,c,j:

c:=0:

for j from 1 to nops(SP) do
	for i from 1 to n do 
	#for j from 1 to nops(SP) do
		if (is({i, i+1} subset SP[j]) = true) then
			c := c+1:
		fi:
	od:
od:

c:
end:


#EstimateAvAdjacentMates(n,K):that uses the above procedure, and your RandSPn(n), K #times, to estimate the average of the number of adjacent mates in a random set partition #of {1,...n}. 

EstimateAvAdjacentMates := proc(n,K) local C,N,i:

C:=0:

for i from 1 to K do
	N:=NuAdjacentMates(RandSPn(n),n):
	C:=C+N:
od:

C/K:
end:


###the code is working but it took so long time without ending for n= 100.
###but it work for EstimateAvAdjacentMates(10,1000), which are 11041/500 , 11083/500  , ### 441/20 , 881/40, 22047/1000