Help: Obtain a huge amount of data solved in FrobeniusSolve? (perhaps better without running FrobeniusSolve?)

Question

How to obtain a huge amount of data solved in FrobeniusSolve more systematically?

I am interested in obtaining these two sets of data from FrobeniusSolve:

sol = FrobeniusSolve[Table[1, {i,1,36}], 8]    -- (command 1)

and

sol = FrobeniusSolve[Table[1, {i,1,70}], 16]  -- (command 2)

But running either of each of these two commands -- I run out of memory and the Mathematica quit its kernel after 3 days or so.

Question: Do you have any way to obtain such a data sol defined above?

What I can do at my best is running

sol = FrobeniusSolve[Table[1, {i,1,20}], 8] -- (command 0)

using Timing[sol = FrobeniusSolve[Table[i, {i,1,20}], 8];]

It shows {508.029, Null} that it costs 508.029 secs, about 8 minutes 20 secs.

For given n and b, I know the meaning of

FrobeniusSolve[Table[1, {i,1,n}], b]

means that solving the set of all possible equations of the Frobenius equation $$ x_1+\dots+x_n=b,$$ where the coefficients all are positive integer 1, while $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-negative integers, i.e. $$ x_j \in \mathbb{N} $$ as Natural numbers. See https://mathoverflow.net/q/352331/27004

It turns out that the number of solutions $f(n,b)$ is given by

$$ f(n,b)= \binom{b+n-1}{n-1}. $$

So (command 0), (command 1) and (command 2) requires these numbers of solutions: $$ f(n=20,b=8)= \binom{8+20-1}{20-1}=2220075. $$ $$ f(n=36,b=8)= \binom{8+36-1}{36-1}=145008513. $$ $$ f(n=70,b=16)= \binom{16+70-1}{70-1}=78682166288559225. $$ So the rough estimation of time to run (command 1) in my computer is: $$ f(n=36,b=8)/f(n=20,b=8) \times 500 sec \simeq 65 \times 500 sec. $$ The rough estimation of time to run (command 2) in my computer is: $$ f(n=70,b=16)/f(n=20,b=8) \times 500 sec \simeq 3.5 \times 10^{10} \times 500 sec. $$ These are literally impossible! (Using a Laptop.)

Are there any simple ways to obtain these data of (command 1) and (command 2) easily? (perhaps without running the commands?) It is also possible to simplify the math equations by reiteration, etc.

See the math: (https://mathoverflow.net/q/352331/27004)

NOTE add: To @yarchik (see comments), NO, I do not just need the number of solutions. I need the long strings of such data as sol above. I need the sol as the basis/bases for solving some nonlinear algebraic equations.

From your post on mathoverflow (MO) I learnt that the goal might be just the number of solutions. If so, FrobeniusSolve is not needed. You can just apply the generating formula from MO. — yarchik, Feb 18 '20 at 07:26
Could you explain what for do you need such results of FrobeniusSolve? There are two posts I could point out as analogous (at least in certain aspects): Finding the number of solutions to a diophantine equation and Finding long strings of identical digits in transcendental numbers — Artes, Feb 18 '20 at 12:31
@yarchik, NO, I do not just need the number of solutions. I need the long strings of such data as sol. — wonderich, Feb 18 '20 at 15:19
From the documentation: "The Frobenius equation is the Diophantine equation $a_1x_1+\ldots+a_n x_n=b$, where the $a_i$ are positive integers, $b$ is an integer, and a solution $x_1,\ldots,x_n$ must consist of non-negative integers." This does not match what you write. — Roman, Feb 18 '20 at 19:14
Sorry, the correction, what I am interested in is: FrobeniusSolve[Table[1, {i,1,n}], b] instead of FrobeniusSolve[Table[i, {i,1,n}], b] — wonderich, Feb 19 '20 at 19:43

score 2 · Answer 1 · answered Feb 18 '20 at 18:41

As long as $n\ge b$, the number of solutions of FrobeniusSolve[Range[n], b] is PartitionsP[b], which is not all that large.

For example, your command 1

FrobeniusSolve[Range[36], 8]

has PartitionsP[8] == 22 solutions:

{{0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,2,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {1,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {1,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {2,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {2,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {3,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {3,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {4,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {4,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {5,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {6,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}}

Your command 2 has 231 solutions:

FrobeniusSolve[Range[70], 16]

{{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 [...]
 {13,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {14,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},
 {16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}}

However, I want to say that $ f(n=20,b=8)= \binom{8+20-1}{20-1}=2220075 $ is a correct statement --- they contain a large set of these solutions. Perhaps you do not consider all the permutations among your answer. — wonderich, Feb 18 '20 at 19:28
@wonderich please show me a solution I didn't consider. Permutations aren't allowed here as the solutions are position-dependent. Are you sure you understand what FrobeniusSolve does? — Roman, Feb 18 '20 at 19:51
I am sure I understand: this is what I need FrobeniusSolve[Table[i, {i,1,n}], b] — wonderich, Feb 18 '20 at 22:57
I need the output of FrobeniusSolve[Table[i, {i,1,36}], 8], FrobeniusSolve[Table[i, {i,1,70}], 16] and FrobeniusSolve[Table[i, {i,1,20}], 8] — wonderich, Feb 18 '20 at 22:58
@wonderich that's precisely what I gave you. I simply used Range[n] instead of your Table[i, {i,1,n}], which is the same thing. I am baffled by your reactions. — Roman, Feb 19 '20 at 06:27
Can you simply run this to know the difference of mine and yours? FrobeniusSolve[Table[i, {i,1,20}], 8] — wonderich, Feb 19 '20 at 19:35
Sorry I was wrong it should be FrobeniusSolve[Table[1, {i,1,20}], 8] — wonderich, Feb 19 '20 at 19:37
I found my typos in the original post: "Table[i, {i,1,n}]" corrected to "Table[1, {i,1,n}]" thanks! — wonderich, Feb 19 '20 at 19:39

Help: Obtain a huge amount of data solved in FrobeniusSolve? (perhaps better without running FrobeniusSolve?)

1 Answers1