#OK to post homework
#Tianyi Liu,Sept 20, Assignment 3
1.
Bnk(10,5)[20]
[0, 0, 0, 1, 1, 1, 1, 0, 1, 0]

MyChoose({1,2,3,4,5,6},2)[5]
{1, 6}

MyPermsL([r,u,t,g,e,r,s])[100]
[r, u, s, t, e, r, g]

WtoS([1,0,0,0,1])
{1, 5}

2.
NuFP:=proc(pi) local nu,i,n:
n:=nops(pi):
nu:=0:
if n=0 then
  RETURN (nu):
fi:
for i from 1 to n do
  if i=pi[i] then
    nu:=nu+1:
  fi:
od:
nu:
end:

3.
Der:=proc(n) local L,i,S,nu:
S:={}:
L:=MyPerms(n):
nu:=nops(L):
for i from 1 to nu do
  if NuFP(L[i])=0 then
    S:=S union {L[i]}:
  fi:
od:
S:
end:

4.
[seq(nops(Der(i)),i=0..8)]
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]

Yes. A000166:Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

5.
[seq(nops(Comps(i)),i=1..8)]
[1, 2, 4, 8, 16, 32, 64, 128]
nops(Comps(i))=2^(i-1)