#OK to post
#Yuxuan Yang, Jan 28th, Assignment 2 

Help:=proc(): print(`SetToVec(S,n),VecToSet(v)`): end:
with(combinat):

#SetToVec(S,n): inputs a subset S of {1,...,n} and outputs the 0-1 vector of length n such that v[i]=1 iff i belongs to S. 
SetToVec:=proc(S,n) local nn,j,i,v:

if not type(S,set) then
print(S, `should be a set `):
 RETURN(FAIL):
fi:

if not (type(n,integer) and n>=0) then
print(n, `should be an integer`):
 RETURN(FAIL):
fi:

nn:={seq(j,j=1..n)}:
if not S subset nn then
print(S, `should be a subset of {1,2,...,n}`):
 RETURN(FAIL):
fi:

v:=[]:
for i from 1 to n do
 if evalb(i in S) then 
  v:=[op(v),1]:
 else
  v:=[op(v),0]:
 fi:
od:
v:
end:

VecToSet:=proc(v) local S,i,n:
S:={}:
n:=nops(v):
for i from 1 to n do
 if v[i]=1 then
  S:=S union {i}:
 fi:
od:
S:
end:

#Box from C2
Box:=proc(L) local k,i,L1,B1,b1:
option remember:

if not type(L,list) then
  print(L, `should be a list `):
 RETURN(FAIL):
fi:

k:=nops(L):
if k=0 then 
 RETURN([[]]):
fi:
 if not ({seq(type(L[i],integer),i=1..nops(L))}={true} and min(op(L))>=0) then
   print(`bad input`):
  RETURN(FAIL):
 fi:

L1:=[op(1..k-1,L)]:

B1:=Box(L1):
 [ seq(seq([op(B1[j]),i],i=0..L[k]),j=1.. nops(B1))]:
end:
#MyPS from C2
MyPS:=proc(S) local s,S1,PS1,s1:

if not type(S,set) then
print(S, `should be a set `):
 RETURN(FAIL):
fi:

if S={} then
 RETURN({{}}):
fi:

s:=S[nops(S)]:

S1:=S minus {s}:

PS1:=MyPS(S1):

PS1 union { seq( s1 union {s}, s1 in PS1)}:

end:

#CheckBij(n): checks that SetToVec(VecToSet(v),n)=v is always true for each member of Box([1$n]) and VecToSet(SetToVec(S,n))=S for each member of MyPS({seq(i,i=1..n)}) .
CheckBij:=proc(n) local S,i,j,nones,B,M:
nones:=[seq(1,i=1..n)]:
B:=Box(nones):
for j from 1 to nops(B) do
 if not SetToVec(VecToSet(B[j]),n)=B[j] then
  RETURN(FAIL):
 fi:
od:
M:=MyPS({seq(i,i=1..n)}):
for j from 1 to nops(M) do
 if not VecToSet(SetToVec(M[j],n))=M[j] then
  RETURN(FAIL):
 fi:
od:
RETURN(TURE):
end:

#NextInLine(L,v):inputs a list L and a member v of the list of lists (dictionary) and returns the vector right after it (in dictionary (lex.) order) without actually constructiong the (usually) very large set Box(L)
NextInLine:=proc(L,v) local v1,n,i,j:
n:=nops(L):
v1:=v:
for j from 1 to n do
 i:=n+1-j:
 if v1[i]=L[i] then 
  v1[i]:=0:
 else
  v1[i]:=v1[i]+1:
  RETURN(v1):
 fi:
od:

print(v,`is a last one`):
end: