How to convert a list of polynomials with different degrees into a list of lists?

Question

I asked a question about converting a polynomial into a list.

Now I have a list of polynomials. For example,

r1={ p[1,2,4], p[1,2,5], p[1,3,4]*p[1,2,5], p[1,3,4]*p[1,2,5]-3*p[1,2,3]*p[2,3,4], p[1,3,4]*p[1,2,5]*p[3,6,9]-3*p[1,2,3]*p[2,3,4]*p[2,5,8], p[1,3,4]*p[1,2,5]*p[3,6,9]-3*p[1,2,3]*p[2,3,4]}

The polynomials in r1 have different degrees and a polynomial can be non-homogeneous. I would like to obtain:

r2={ {{1, {{1,2,4}}}}, {{1, {{1,2,5}}}}, {{1, {{1,3,4},{1,2,5}}}}, {{1,{{1,3,4},{1,2,5}}}, {-3, {{1,2,3},{2,3,4}}}}, { {1,{{1,3,4},{1,2,5},{3,6,9}}}, { -3, {{1,2,3},{2,3,4},{2,5,8}} } }, { {1,{{1,3,4},{1,2,5},{3,6,9}}}, { -3, {{1,2,3},{2,3,4}} } } }

How to obtain it? Thank you very much.

Answer 1

This gets a bit complicated because MMA will automatically evaluate addition of structurally equal elements. Toward this aim, we change addition (subtraction is an addition of the negative element) to a list and mark it by: "q":

rule0 = c1_ + c2_ -> {q, {c1, c2}, q};

In the end, we will undo this.

Then we write rules for changing p[...] and product thereof to lists:

rule1 = (c1_Integer : 1)  p[c2__] -> {{c1, {{c2}}}};
rule2 = (c1_Integer : 1)  p[c2__]  p[c3__] -> {c1, {{c2}, {c3}}};
rule3 = (c1_Integer : 1) p[c2__]  p[c3__]  p[ c4__] -> {c1, {{c2}, {c3}, {c4}}} ;

Finally, we need to undo the "{q" and "q}":

rule4 = {q, c__, q} -> c;

With these rules we may now transform r1:

 r1 /. rule0 /. {rule3, rule2, rule1}   /. rule4

{{{1, {{1, 2, 4}}}}, {{1, {{1, 2, 5}}}}, {1, {{1, 2, 5}, {1, 3, 
    4}}}, {{1, {{1, 2, 5}, {1, 3, 4}}}, {-3, {{1, 2, 3}, {2, 3, 
     4}}}}, {{-3, {{1, 2, 3}, {2, 3, 4}, {2, 5, 8}}}, {1, {{1, 2, 
     5}, {1, 3, 4}, {3, 6, 9}}}}, {{-3, {{1, 2, 3}, {2, 3, 
     4}}}, {1, {{1, 2, 5}, {1, 3, 4}, {3, 6, 9}}}}}