#its ok to post
#TaerimKim,12/13/2020,Assignment 24


#1. use the functions for each cases
#(i)
nops(PnC(97, {1, 4, 7}, 7));
                             10950
#(ii)
nops(PnD(97,3));
			19990


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#2. combine pnkC + pnkD for pnkW and call from pnkDC

#pnkW(n,k,C,m,d): 
pnkW:=proc(n,k,C,m,d) local k1,S:
option remember:
if k>n then
 RETURN(0):
fi: #same for pnkC and pnkD

if k=n then
 if member(k mod m, C) then
 RETURN(1):
 else
 RETURN(0):
 fi:
fi: #must satisfy both so this condition remains

if n=1 then
 if member(1,C) and k=1 then
   RETURN(1):
 else
  RETURN(0):
 fi:
fi: #combined condition

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+ pnkW(n-k,k1,C,m,d):
fi:

od:
S:
end:

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

#(i)
#lets check with n=10,C={1},m=2
pnDC(10,0,{1},2)
10
pnC(10,{1},2)
10
evalb(%=%%)
true

#lets check with n=10,d=1
pnDC(10,1,{0},1)
10
pnD(10,1)
10
evalb(%=%%)
true

#so its ok

#(ii). when it is odd and distinct we can use d=1, C={1}, m=2
seq(pnDC(i,1,{1},2),i=1..40);
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,Expansion of Product_{k>=0} (1 + x^(2k+1)); number of partitions of n into distinct odd parts;
#seems good

#(iii). this condition asks for d=2, C={1,4}, m=5
seq(pnDC(i,2,{1,4},5),i=1..40);
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,Number of partitions of n into distinct parts 5k+1 or 5k+4.

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#3. For distinct partitions given that frequency notation is as 1^a1 2^a2 3^a3 so forth
#we know that for each numbers in distinct case a1,a2,a3,...,ak are all 1
#meaning there is only {[].[1]} for 1,{[].[2]} for 2, ... ,{[].[k]} for k
#SO for |PAR|_q = |PAR(1's) x Par(2's) x ... x Par(k's)|_q = | PAR(1's)|_q + |PAR(2's)|_q ... 
#=(1+q)*(1+q^2)*(1+q^3)*...*(1+q^k)
#=Product(1+q^i,i=1..infinity) Q.E.D


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#4. For Odd partitions we know that numbers of 2n are not included 
#meaning there are no cases for a(2n)
#So {[].[1],[11],...[1$k]} for 1,(no case for 2), {[].[3],[33],...[3$k]} for 3, ... ,{[].[k],[kk],...[k$k]}for k=2n+1 from n=0..infin
#Then our |Par|_q = (1+q+q^2+q^3+....) * (1+q^3+q^6+...) * (1+q^5+q^10+...)* ... 
#= 1/(1-q) * 1/(1-q^3) * 1/(1-q^5) * ....
#= Product(1/(1-q^(2*i+1),i=0..infinity) Q.E.D

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#5. We will utilize a weapon that shows (1+a^i) = (1-a^2i)/(1-a^i)
#Then from distinct GF
#(1+q)*(1+q^2)*(1+q^3)*...*(1+q^k) = (1-q^2)/(1-q) * (1-q^4)/(1-q^2) * (1-q^6)/(1-q^3) * .... *
#here we can see that the (1-q^2n) from denom. and num. are cancelled out. leaving (1-q^2n-1) in the denom.
#= 1/((1-q)*(1-q^3)*(1-q^5)*...)
#=Product(1/(1-q^(2*i+1),i=0..infinity) = Odd partition GF Q.E.D