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There is a marvelous combinatorial class called IntegerListsLex that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try trying IntegerListsLex? at the Sage command line to see the documentation.

Here is an iterator that I think does the job:

sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[9]*7)
sage: for i in L:
....:     print i
....:
[6, 1, 1, 1, 1, 1, 1]
[5, 2, 1, 1, 1, 1, 1]
[5, 1, 2, 1, 1, 1, 1]
[5, 1, 1, 2, 1, 1, 1]


etc...

There is a marvelous combinatorial class called IntegerListsLex that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try trying entering IntegerListsLex? at the Sage command line to see the documentation.

Here is an iterator that I think does the job:

sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[9]*7)
sage: for i in L:
....:     print i
....:
[6, 1, 1, 1, 1, 1, 1]
[5, 2, 1, 1, 1, 1, 1]
[5, 1, 2, 1, 1, 1, 1]
[5, 1, 1, 2, 1, 1, 1]


etc...

There is a marvelous combinatorial class called IntegerListsLex that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try entering IntegerListsLex? at the Sage command line to see the documentation.

Here is an iterator that I think does the job:

sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[9]*7)
sage: for i in L:
....:     print i
....:
[6, 1, 1, 1, 1, 1, 1]
[5, 2, 1, 1, 1, 1, 1]
[5, 1, 2, 1, 1, 1, 1]
[5, 1, 1, 2, 1, 1, 1]


etc...

The Lex part of the name indicates that the lists are returned in lexicographic order.

There is a marvelous combinatorial class called IntegerListsLex that returns an iterator (a type of python object that you can iterate over) for lists of integers subject to certain constraints. Try entering IntegerListsLex? at the Sage command line to see the documentation.

Here Below is an iterator that I think does the job:job. The length parameter limits to sequences of length 7, the floor parameter limits the integers to be larger or equal to 1, the ceiling parameter limits the integers to be less or equal to 3.

sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[9]*7)
ceiling=[3]*7)
sage: for i l in L:
....:     print i
l
....:
[6, 1, 1,
[3, 3, 2, 1, 1, 1, 1]
[5, 2, 1, 1, [3, 3, 1, 2, 1, 1, 1]
[5, 1, 2, 1, 1, [3, 3, 1, 1, 2, 1, 1]
[5, 1, 1, 2, 1, 1, [3, 3, 1, 1, 1, 2, 1]


etc...

The Lex part of the name indicates that the lists are returned in lexicographic order.

Note that if you sort the sequences, convert to tuples (so that they are hashable), and then uniquify them you get the 3 basic sequences that generate the whole collection by reordering:

sage: L = IntegerListsLex(12, length=7, floor=[1]*7, ceiling=[3]*7)
sage: LS = [ tuple(sorted(i)) for i in L ]
sage: uniq(LS)
[(1, 1, 1, 1, 2, 3, 3), (1, 1, 1, 2, 2, 2, 3), (1, 1, 2, 2, 2, 2, 2)]


Update changed ceiling option to limit the sequence to 1's, 2's, and 3's