> #Weiji Zheng, 10/03/2020, Assignment 8
;
> #OK to post homework
;
> 
;
> #Q1
;
> 
;
> #Code
;
> CompsGFk:=proc(S,x,k) local s:
> sort(expand((add(x^s, s in S))^k)):
> end:
> CompsGF:=proc(S,x) local s:
> if not (type(S,set) and min(op(S))>0 ) then
>  print(`Bad input`):
>  RETURN(FAIL):
> fi:
> 
> factor(1/(1-add(x^s,s in S))):
> 
> end:
> CompsGFk({1,2,3},x,2);
x^6+2*x^5+3*x^4+2*x^3+x^2
;
> #CompsGFk(S,x,k): inputs a FINITE set of INTEGERS (0 and negative integers are OK!) and a variable x, and
> #a non-negative integer k, outputs the POLYNOMIAL (possibly Laurent polynomial, i.e. negative powers OK)
> #whose coefficient of x^n is the EXACT number of words of length k, in the finite alphabet S
> #that add-up to n. For example to get the generating function of all 4-letter words in {0,1} according to the
> #sum (alias "number of 1s") try:
>  
;
> #The number of 10-letter words in the alphabet {1,4,6} that add-up to 201
;
> CompsGFk({1,4,6},x,10)
x^60+10*x^58+45*x^56+10*x^55+120*x^54+90*x^53+210*x^52+360*x^51+297*x^50+840*x^49+570*x^48+1260*x^47+1380*x^46+1380*x^45+2565*x^44+1680*x^43+3160*x^42+2880*x^41+2731*x^40+4290*x^39+2520*x^38+4210*x^37+3510*x^36+2772*x^35+4245*x^34+2100*x^33+3150*x^32+2640*x^31+1470*x^30+2520*x^29+1050*x^28+1260*x^27+1260*x^26+372*x^25+840*x^24+360*x^23+210*x^22+360*x^21+45*x^20+120*x^19+90*x^18+45*x^16+10*x^15+10*x^13+x^10
;
> #No number of 10-letter words in the alphabet {1,4,6} can add-up to 201
;
> 
;
> # The number of words in the alphabet {2,3,7} of ANY length, that add-up to 411
;
> # It's gonna be infinity of any length
;
> 
;
> # The number of sequenes of any length whose entries are either 1 (mod 5) or 4 (mod 5) and that add up to 341
;
> # 1 (mod 5) = 1
;
> # 4 (mod 5) = 4
;
> CompsGF({1,4},x)
-1/(x^4+x-1)
;
> #I think the number is still infinity.
;
> 
;
> #Q2
;
> 
;
> with(combinat,partition)
[partition]
;
> with(ListTools):
> Pnk := proc(n,k) local P,a,b,S:
> option remember:
> 
> P := partition(n,k):
> S := {}:
> 
> for a from 1 to nops(P) do
> 
>     if FindMaximalElement(P[a])=k then
> 
>         S:=S union {sort(P[a],`>`)}:
>     fi:
> od:
> 
> S:
> end:
> #TEST
;
> Pnk(5,2)
{[2, 2, 1], [2, 1, 1, 1]}
;
> 
> #Q3
> 
;
> pnk := proc(n,k) local num,SS:
> option remember:
> 
> SS := Pnk(n,k):
> num :=numelems(SS):
> num:
> end:

> pnk(5,3)
2
;
> pnk(5,4)
1
;
> pnk(5,2)
2
;
> 
;
> #Q4
;
> 
;
> pn := proc(n) local i,SSS,res:
> option remember:
> 
> SSS:={}:
> 
> if n=0 then
>    RETURN(1):
> else
>    for i from 1 to n do
>        SSS:=SSS union Pnk(n,i):
>    od:
> fi:
> res := numelems(SSS):
> res:
> end:
> 
;
> pn(5)
7
;
> pn(0)
1
;
> seq(pn(n),n=0...30);
1, 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, denoting "a(n) is the number of partitions of n (the partition numbers)."
;
> 
;
> #Q5
;
> 
;
> [seq(p(5*n+4) mod 5, n=1..50)]
[p(9), p(14), p(19), p(24), p(29), p(34), p(39), p(44), p(49), p(54), p(59), p(64), p(69), p(74), p(79), p(84), p(89), p(94), p(99), p(104), p(109), p(114), p(119), p(124), p(129), p(134), p(139), p(144), p(149), p(154), p(159), p(164), p(169), p(174), p(179), p(184), p(189), p(194), p(199), p(204), p(209), p(214), p(219), p(224), p(229), p(234), p(239), p(244), p(249), p(254)]
;
> [seq(p(7*n+5) mod 7, n=1..50)]
[p(12), p(19), p(26), p(33), p(40), p(47), p(54), p(61), p(68), p(75), p(82), p(89), p(96), p(103), p(110), p(117), p(124), p(131), p(138), p(145), p(152), p(159), p(166), p(173), p(180), p(187), p(194), p(201), p(208), p(215), p(222), p(229), p(236), p(243), p(250), p(257), p(264), p(271), p(278), p(285), p(292), p(299), p(306), p(313), p(320), p(327), p(334), p(341), p(348), p(355)]
;
> [seq(p(11*n+6) mod 11, n=1..50)]
[p(17), p(28), p(39), p(50), p(61), p(72), p(83), p(94), p(105), p(116), p(127), p(138), p(149), p(160), p(171), p(182), p(193), p(204), p(215), p(226), p(237), p(248), p(259), p(270), p(281), p(292), p(303), p(314), p(325), p(336), p(347), p(358), p(369), p(380), p(391), p(402), p(413), p(424), p(435), p(446), p(457), p(468), p(479), p(490), p(501), p(512), p(523), p(534), p(545), p(556)]
;
> #it takes too much time to computing
;