#OK to post homework 
#Zhizhang Deng, 10/01/2020, Assignment 8

1. 
(i). 0 
(ii). 4741685087960650685822461715671211099459856575685906645393
(iii). 234303065886413369536085349033389448858104244347193374824719

2. 
Pnk := proc(n, k) local res, i, tmpres, idx;
option remember; 

if k > n then
    return {};
fi; 

if k = 1 and n = 1 then
    return {[1]}
fi;

if k = n then 
    return {[k]}
fi; 

res := {};

for i from k by -1 to 1 do
    tmpres := Pnk(n - k, i);
    res := res union {seq([k, op(tmpres[idx])], idx=1..nops(tmpres))}
end do;

return res;

end:

3. 
pnk := proc(n, k) local res, i ;

if k > n then
    return 0;
fi; 

if k = 1 and n = 1 then
    return 1
end if;

if k = n then
    return 1
end if;

res := 0;

for i from k by -1 to 1 do
    res := res + pnk(n - k, i);
end do;

return res; 

end: 

4. 
pn := proc(n) local res, i, k;

option remember; 

res := 0;

for k from n by -1 to 1 do 

    if k > n then
        next;
    fi; 

    if k = 1 and n = 1 then
        res := res + 1; 
        next; 
    end if;

    if k = n then
        res := res + 1;
        next; 
    end if;

    for i from k by -1 to 1 do
        res := res + pnk(n - k, i);
    end do;

end do; 

end: 

seq(pn(n), n=0..30) = 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604
A-Number is A000041

5. 
All of them are 0