Solving Sum of Integers for Given Constraints

[TriTertButoxy]

Hi, I'm doing a physics calculation, and along the way, I've run up against a curious math problem. I'm sure this is a rather classic problem in mathematics, but I'm just not acquainted with the subject enough to answer it, or even look it up, so hopefully someone can point me in the right direction.

For a given positive integer $\ell>0$ and another positive integer $m\leq\ell$, for what values of a list of (zero or positive) integers $\{k_1\geq0,\,k_2\geq0,\,\ldots,\,k_\ell\geq0\}$ satisfies the following simultaneous pair of equations?
$$\sum_{n=1}^\ell k_n = m$$
$$\sum_{n=1}^\ell n k_n = \ell$$

Even the name given to the problem would point me in the right direction. Thanks!
Also, please do not treat me like a student :-)

 

[TriTertButoxy]

Ok, I think I found the answer; The answer has something to do with Bell's polynomials, and exponential formula. Thanks anyway :-)

 

[ramsey2879]

Hi, I'm doing a physics calculation, and along the way, I've run up against a curious math problem. I'm sure this is a rather classic problem in mathematics, but I'm just not acquainted with the subject enough to answer it, or even look it up, so hopefully someone can point me in the right direction.

For a given positive integer $\ell>0$ and another positive integer $m\leq\ell$, for what values of a list of (zero or positive) integers $\{k_1\geq0,\,k_2\geq0,\,\ldots,\,k_\ell\geq0\}$ satisfies the following simultaneous pair of equations?
$$\sum_{n=1}^\ell k_n = m$$
$$\sum_{n=1}^\ell n k_n = \ell$$

Even the name given to the problem would point me in the right direction. Thanks!
Also, please do not treat me like a student :-)

For the sum to add up to no more than $\ell$ there would have to be a lot of k's equal to zero since for larger integers the sum would easily exceed $\ell$. To find a solution, I would suggest that assume all but one k were zero and that m divides $\ell$. P.S. this doesn't look like a classic problem.