How to count all restricted partitions of the number $155$ into a sum of $10$ natural numbers between $[0,30]$?

Question

I really have no clue what to do with this one. Thanks for any help!

(+1) I only know how to solve this without the restriction that the numbers are between $[0,30]$ would like to know the solution for this too. — Yanko, Dec 07 '17 at 18:49
Order doesn't matter. I would rather see a mathematical solution, but any kind of help will be appreciated! — Herrho, Dec 07 '17 at 18:54
So $154+1+0+0+\cdots$ and $1+154+0+0+\cdots$ are counted as one? — robjohn, Dec 07 '17 at 20:00
Even though neither of them would count since they don't meet the restriction, but you get the idea. — robjohn, Dec 07 '17 at 20:13
@Hewwho: since you are speaking of Partitions then order does not matter, otherwise you would speak of Compositions — G Cab, Dec 07 '17 at 22:14

score 1 · Answer 1 · answered Dec 07 '17 at 19:11

We can find the number of restricted partitions of $n$ into $k$ elements in $\{0,1,\dots, m\}$ in time $nkm$ using a $2$-d knapsack.

To do this let $A[i][j]$ be the number of resitricted partitions of $i$ into $j$ summands and actualize $A$ to incorporate each of the $m+1$ numbers one by one.

Some code:

#include <bits/stdc++.h>
using namespace std;

const int MAX=200;
int A[MAX][MAX]; // the array

int main(){
    int N=155,K=10,M=30;
    for(int i=0;i<=K;i++){ // set it up so that the number 0 can be formed in 1 way for any number of summands
        A[0][i]=1;
    }
    for(int i=1;i<=M;i++){ // incorporate the number i as a possible summand
        for(int j=i;j<=N;j++){ // actualize the number of ways to form the number j starting from j=i
            for(int l=1;l<=K;l++){ 
                A[j][l]+=A[j-i][l-1]; // the number of ways to make j with l summands increases by A[j-i][l-1]
            }
        }
    }
    printf("%d\n",A[N][K]);
}

output is $10270579$

rtybase · Answer 2 · 2017-12-07T23:16:17.473

So, formalised problem is $$\sum\limits_{k=1}^{10} x_k=155$$ $$0\leq x_k\leq 30, \forall k=1..30$$ Using the following generating function (here is an example also pointing to other examples and a book) $$(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+\\x^{10}+x^{11}+x^{12}+x^{13}+x^{14}+x^{15}+x^{16}+x^{17}+x^{18}+x^{19}+\\x^{20}+x^{21}+x^{22}+x^{23}+x^{24}+x^{25}+x^{26}+x^{27}+x^{28}+x^{29}+x^{30})^{10}$$ and a computer (to avoid tedious calculations) to find the coefficient of the $x^{155}$ term, the answer is $11219423909980$.

But this question isn't totally clear, as per the question Rob raised in the comments and 2 closing votes. It turns out, there is a definition of restricted partitions as defined here. That link also states the generating function as Theorem 3.1 (same result here, as $(3.9)$).

These are Compositions not Partitions ! – G Cab Dec 07 '17 at 22:04 — G Cab, Dec 07 '17 at 22:04
Rob's question wasn't answered yet though – rtybase Dec 07 '17 at 22:08 — rtybase, Dec 07 '17 at 22:08

score 1 · Answer 3 · answered Dec 07 '17 at 22:07

So you are looking for $$ N_p (s,r,m) = {\rm No}{\rm .}\,{\rm of}\,{\rm integer}\;{\rm solutions}\;{\rm of}\left\{ \matrix{ 0 \le x_{\,j} \le r \hfill \cr x_{\,j} \le x_{\,j + 1} \hfill \cr \sum\limits_{1\, \le \,j\, \le \,m} {x_{\,j} } = s \hfill \cr} \right. $$

Now, while the number of restricted Compositions $$N_{\,b} (s,r,m) = \text{No}\text{. of solutions to}\;\left\{ \begin{gathered} 0 \leqslant \text{integer }x_{\,j} \leqslant r \hfill \\ x_{\,1} + x_{\,2} + \cdots + x_{\,m} = s \hfill \\ \end{gathered} \right.$$ can be expressed by $$ N_b (s,r,m)\quad \left| {\;0 \leqslant \text{integers }s,m,r} \right.\quad = \sum\limits_{\left( {0\, \leqslant } \right)\,\,k\,\,\left( { \leqslant \,\frac{s} {r}\, \leqslant \,m} \right)} {\left( { - 1} \right)^k \left( \begin{gathered} m \hfill \\ k \hfill \\ \end{gathered} \right)\left( \begin{gathered} s + m - 1 - k\left( {r + 1} \right) \\ s - k\left( {r + 1} \right) \\ \end{gathered} \right)} $$ as explained in this other post, and $Nb(155,30,10)$ gives the value provided by rtybase.

I also have been looking for an expression for the Partitions, but did not succeed in that.

How to count all restricted partitions of the number $155$ into a sum of $10$ natural numbers between $[0,30]$?

3 Answers3