#OK to post homework
#Tianyi Liu, Dec13, Assignment 24
1.
i.108236
ii.19990

2.
pnkDC:=proc(n,d,k,C,m) local k1,S:
option remember:
if k>n then
 RETURN(0):
fi:

if k=n then
 if member(k mod m, C) then
 RETURN(1):
 else
 RETURN(0):
 fi:
fi:

if n=1 then
 if member(1,C) then
  
   RETURN(1):
 else
  RETURN(0):
 fi:
fi:


if not member(k mod m, C) then
 RETURN(0):
fi:


S:=0:
for k1 from 1 to k-d do

if member(k1 mod m,C) then
 S:=S+ pnkDC(n-k,d,k1,C,m):
fi:

od:
S:
end:
pnDC:=proc(n,d,C,m) local k:
add(pnkDC(n,d,k,C,m),k=1..n):
end:

i.
pnDC(10,0,[2,3],5)
4
pnC(10,[2,3],5)
4
pnDC(10,2,[0],1) 
6
pnD(10,2)
6

ii.
1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 8, 9, 11, 12, 12, 14, 16, 17, 18, 20, 23, 25, 26, 29, 33, 35, 37, 41, 46
A000700

iii.
1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 5, 3, 3, 5, 7, 7, 6, 5, 7, 11, 11, 8, 8, 12, 15, 15, 13, 12, 16, 22
A203776


3.
We use similar argument for the full partition. When we pick a random integer partition we make decisions that how many number
to include. But since distinct partition must have all distinct parts in a partition, we can only decide to include a number or
not. Then
PAR=PAR(ONLY USING 1)*PAR(ONLY USING 2)*...
{[],[1]}*{[],[2]}*...
(1+q)*(1+q^2)*...=Product(1+q^i,i=1..infinity)

4.
We use similar argument for the full partition. When we pick a random integer partition we make decisions that how many number
to include. But since odd partition must have only odd parts in a partition, we can only decide to include an odd number or
not. Then
PAR=PAR(ONLY USING 1)*PAR(ONLY USING 3)*...
{[],[1],[1,1]...}*{[],[3],[3,3]...}*...
1/((1-q)*(1-q^3)*(1-q^5)*....)=Product(1/(1-q^(2*i+1),i=0..infinity)

5.
1/((1-q)*(1-q^3)*(1-q^5)*....)=Product(1/(1-q^(2*i+1),i=0..infinity)
=(1+q)/(1-q)(1+q)*(1+q^3)/(1-q^3)(1+q^3)*(1+q^5)/(1-q^5)(1+q^5)*....
=(1+q)(1+q^2)/(1-q^2)(1+q^2)...
=(1+q)(1+q^2)/(1-q^4)*(1+q^3)/(1-q^3)(1+q^3)...
=(1+q)(1+q^2)(1+q^3)/(1-q^4)(1-q^6)...
=(1+q)(1+q^2)(1+q^3)(1+q^4)/(1-q^8)(1-q^6)...
=(1+q)*(1+q^2)*...=Product(1+q^i,i=1..infinity)