OrderedSetPartitions (variant)

edit

I would like to have the list of partitions of [1,2,...N] of finite size k but with using the NULL set

for example, if looking for the partitions of [1,2,3,4,5] of size 3, i would like [ [],[],[1,2,3,4,5] ]

and [ [],[1,2,3],[4,5] ] to be in this list

is it possible to do natively in Sage, or should i write a custom function ?

OrderedSetPartitions(5,3).list()

returns

[[{1, 2, 3}, {4}, {5}],
 [{1, 2, 3}, {5}, {4}],
 [{1, 2, 4}, {3}, {5}],
 [{1, 2, 5}, {3}, {4}],...

EDIT: here is a (non optimized at all) attempt to solve naively this problem

def OrderedSetPartitions_0(A,k):

    cols={i for i in range(k)}
    # returns the list of k-OrderedSetPartitions of A, allowing for the empty set
    s=Subsets(cols).list()
    res=[]
    count=0
    P=[OrderedSetPartitions(A,i) for i in range(k+1)]

    for sub in s:
           print("sub=")
           print(sub)

           tmp=[ {} for i in range(k)]
           c=sub.cardinality()
           for part in P[c]:
               print("part=")
               print(part)
               for i in range(c):
                   tmp[sub[i]]=part[i]

               print("tmp=")
               print(tmp)

               res=res.append([tmp])
               #  res=res.append([tmp]) # tried this too
               print("res=")
               print(res)
               count=count+1
    return(res)
    # print(count)

A=range(3)
k=2
A
P=[OrderedSetPartitions(A,i) for i in range(k+1)]
# note that P[2].list is a list of list !
P[2].list()

[[{0, 1}, {2}],
 [{0, 2}, {1}],
 [{1, 2}, {0}],
 [{0}, {1, 2}],
 [{1}, {0, 2}],
 [{2}, {0, 1}]]

myset=OrderedSetPartitions_0(A,k)

I get this error message, and I admit I don't get it at all, because it looks fine when coding, but somehow res seems to be "None" instead of []

sub=
{}
sub=
{0}
part=
[{0, 1, 2}]
tmp=
[{0, 1, 2}, {}]
res=
None
sub=
{1}
part=
[{0, 1, 2}]
tmp=
[{}, {0, 1, 2}]
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_21.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("#

-- coding: utf-8 --\n" + _support_.preparse_worksheet_cell(base64.b64decode("bXlzZXQ9T3JkZXJlZFNldFBhcnRpdGlvbnNfMChBLGsp"),globals())+"\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module>

  File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpryfYOj/___code___.py", line 2, in <module>
    exec compile(u'myset=OrderedSetPartitions_0(A,k)
  File "", line 1, in <module>

  File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpSH_9LF/___code___.py", line 27, in OrderedSetPartitions_0
    res=res.append([tmp])
AttributeError: 'NoneType' object has no attribute 'append'

The problem is about res if i omit the line res=res.append(tmp), i will get the enumeration ok i think

EDIT: i changed res=res.append(tmp) by res.append(tmp) i also changed the line initializing tmp into tmp=[ set() for i in range(k)] now as a result of print(tmp) i get the correct enumeration

[{0, 1, 2}, set([]), set([])]
[set([]), {0, 1, 2}, set([])]
[set([]), set([]), {0, 1, 2}]
[{0, 1}, {2}, set([])]
[{0, 2}, {1}, set([])]
[{1 ...