fundamental principles of counting - 2nd try

dear group,

coming back from a break I found no reaction to my query:

|But I don't find functions to list (not only count) all
|. arrangements of s (whose number is n^r )
||. combinations of s (whose number is C(n,r) )
|. variations of s   (:= all combs with repetitons,
|  there are C(n+r-1,r) many of them and we want to see them all)

Maybe I was not explicit enough ..
but I would like to do elementary discrete math with Maxima.

So I will elaborate a bit, give a partial answer
and ask for a solution to concrete question.

Q1: If S is a set, how to produce a sequence of k instances
    of S e.g. to call cartesian_product(S,S,..S)?
    makelist(S,i,1,k) don't work ..

(%i223) S : {1,2,3,4}$
        S2 : cartesian_product (S, S);

Q2: do you see possibilities to 'optimise' my code
    for the 4 cases of counting? see below

Q3: how to write a better version of Events_nOwR(n,k), see below?

HERE is my code.
Any hints are very welcome,

hth Wolfgang

------------------------------------------------------------
Fundamental principles of Counting with MAXIMA: (W. Lindner)
------------------------------------------------------------

1. LISTING OF ALL ELEMENTAY EVENTS WITH MAXIMA.

According to the attachment we have 4 cases.

case 1: nOnR (no order, no repetitions)

(%i269) Events_nOnR(n,r) := full_listify(powerset
(setify(makelist(i,i,1,n)), n-r))$
        Events_nOnR(4,2);
(%o270) [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]

case 2: wOwR (with order, with repetitions)
Problem: how to write function Events_wOwR(n,r),
         e.g. how to produce a sequence of k instances of S
         to call cartesian_product(S,S,..,S) ??

(%i223) S : {1,2,3,4}$
        S2 : cartesian_product (S, S);
        listify(S2);
        cardinality(S2);

(%o224)
{[1,1],[1,2],[1,3],[1,4],[2,1],[2,2],[2,3],[2,4],[3,1],[3,2],[3,3],[3,4],[4,
1],[4,2],[4,3],[4,4]}
(%o225)
[[1,1],[1,2],[1,3],[1,4],[2,1],[2,2],[2,3],[2,4],[3,1],[3,2],[3,3],[3,4],[4,
1],[4,2],[4,3],[4,4]]
(%o226) 16

case 3: wOnR (with order, no repetitions)
problem: .. using result of case 2 ..

(%i273) Events_wOnR(n,r) := listify(subset (S2, lambda ([x], is(
cardinality(setify(x))=k))))$
        Events_nOnR(4,2);

(%o274) [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]

case 4: nOwR (no order, with repetitions)
        we have 10 elementary events:
ElEr: 1 1 [1,1] (ElEr = elementary event)
      1 2
      1 3
      1 4
      2 2
      ..
      3 4
      4 4 -> length=10

(%i281) Events_nOwR (n,k) := block(
ElEr : makelist([i],i,1,n),
for j:2 thru k do (
  ElErNew: [],
  for m:1 thru length(ElEr) do
    for r : ElEr[m][j-1] thru n do
        ElErNew: append( ElErNew, [append(ElEr[m],[r])] ),
  ElEr : ElErNew
  ),
  sublist(ElEr, lambda ([x], x[1]));


(%i282) Events_nOwR(4,2);
(%o282) [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]

2. URN-MODELLS:

Urnmodell nOwR, one draw (with repetions and no order)
Comment: 'sort' destroys the distinguish steps of the drawing!

(%i254)  nOwR(n,r) := sort(makelist(random(n)+1, k,1,n-r))$
         nOwR(4,2);
(%o248) [1,4]

Urnmodell nOnR, one draw:

(%i255) nOnR(n,r) := sort(rest (random_permutation (makelist(i,i,1,n)),
n-r))$
        nOnR(4,2);
(%o248) [1,4]

Urnmodell wOwR, one draw:

(%i263) wOwR(n,r) := makelist(random(n)+1, k,1,r)$
        wOwR(4,2);
(%o256) [2,3]

Urnmodell wOnR, one draw:

(%i265) wOnR(n,r) := rest (random_permutation (makelist(i,i,1,n)), n-r)$
        wOnR(4,2);
(%o264) [3,1]