# Please do not post homework
# Ravali Bommanaboina, 9/13/2020, HW1

# Question 1:-------------------------------------------------------------

#read "C:/Users/rb1002/Documents/hw1.txt";
#nops(F(6));
#                               13
#NuF(6);
#                               13
#nops(F(10));
#                               89
#NuF(10);
#                               89
#diff(x^2, x);
#                              2 x
#


# Question 2:-------------------------------------------------------------
# F(6)={[1,1,1,1,1,1],[2,1,1,1,1],[1,2,1,1,1],[1,1,2,1,1],[1,1,1,2,1],[1,1,1,1,2],[2,2,1,1],[2,1,2,1],[2,1,1,2],[1,1,2,2],[1,2,1,2],[1,2,2,1],[2,2,2]}

# Question 3:-------------------------------------------------------------

#The function L->[op(L),a] appends a new element a to the list L.
#if L1:=[Mercury, Venus, Earth] and we want L2:=[Mercury, Venus, Earth, Mars]
#Then we can run this operation L2:=[op(L1),Mars];
#We initialize WALKS(m,n) to get all the paths from the coordinate (0,0) to the destination (m,n).
# m and n are the number of steps in a particular direction.
# If m represented steps north and n represented steps east then WALKS(5,4) would give us all the paths that eventually go 5 steps north and 4 steps east.
# WALKS(0,0) gives us the empty set because there are 0 steps north and east so there are no paths to the destinations because your starting location is already the destination.

# Question 4:-------------------------------------------------------------
with(combinat):
F:=proc(S) local a,S1,P1,s,numTwos,k,i:
option remember:

P1:={};
numTwos:=floor(S/2);

if S=0 then
 RETURN({{}}):
fi:
a:=0;
while a<=numTwos do
S1:={};
i:=0;
k:=0;
	while k < a do
	S1:=[op(S1),2];
	i:=i+2;
	k:=k+1;
	end do;

	while i<S do
	i:=i+1;
	S1:=[op(S1),1];
	end do;

P1:=[op(P1),op(permute(S1))];
a:=a+1;
end do;

RETURN(convert(P1,set));
end:

#the output for F(6) is the same as what I computed by hand in question 2.
# Question 5:-------------------------------------------------------------

NuF:=proc(S) local a,S1,P1,s,numTwos,k,i:
option remember:
numTwos:=floor(S/2);
s:=0;
a:=0;
k:=0;
while a<=numTwos do
s:=((a+S-k)!/(a!*(S-k)!)) + s;
a:=a+1;
k:=k+2;
end do;

RETURN(s);
end:

#NuF(1000)=70330367711422815821835254877183549770181269836358732742604905087154537118196933579742249494562611733487750449241765991088186363265450223647106012053374121273867339111198139373125598767690091902245245323403501
#Maple would not allow you to compute nops(F(100)) because this function would have to compute all the sets first then count them. This would take too long to compute.