Defining vector partition functions in sage

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• The line p(x) = [] attempts to construct a symbolic function, and when you call par(v1+v2), it calls p(v1+v2) which tries to construct a symbolic function without the correct form for the arguments. I think a dictionary would be better for this.

• Is p supposed to be local to the function, or are its values supposed to be remembered from one call to the next?

• You define p(x) = [], and then you never provide a way to change this: the main loop says for z in p(x-y), but since p(x-y) starts off empty, the loop will never do anything.

EDIT:

You need a place to start: p(x) can't be empty for every x. You also need to know when you're done. Here is a start, not sure if it works:

p = {}

def initialize_p():
    for x in Pos:
        if x:
            # The nonzero vectors should start with the trivial partition.
            p[x] = [[x]]
        else:
            # if x is zero, don't include it in the partition.
            p[x] = []

def par(x):
    for y in S:
        if x-y in Pos:
            for z in p[x-y]:
                # z should be a list, so add a new element to that list
                n = z + [y]
                # sort n so we don't get both [v1, v2] and [v2, v1]
                n = sorted(n, key=str)
                if n not in p[x]:
                    p[x].append(n)
    return p[x]

I put this in a file called vec.sage and did:

sage: var('v1, v2')
sage: S = [v1, v2] # why include v1+v2?
sage: Pos = [k*v1+l*v2 for k in [0,1,2] for l in [0,1,2]]

sage: %attach vec.sage
sage: initialize_p()
sage: par(v1)                                                                                                                
[[v1]]
sage: par(v1+v2)                                                                                                             
[[v1 + v2], [v1, v2]]
sage: par(2*v1+v2)                                                                                                           
[[2*v1 + v2], [v1, v1 + v2], [v1, v1, v2], [2*v1, v2]]