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In a Faa Di Bruno Formula there is an equation:

$m_1$+2*$m_2$+3*$m_3$+...n*$m_n$=n

Is there any general solution for this equation.

For example for $m_1$+$m_2$+$m_3$+...+$m_n$=n, there is a simple algorithm calculating this. Thanks, Gevorg.

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3 Answers 3

I presume you want to solve this equation $\sum_{k=1}^{n}km_k=n$ for integer $m_k$ that sum to $s$. There is no general solution, valid for any $1\leq s\leq n$, but the problem can be reduced to a calculation of integer partitions, see Faa di Bruno's formula, lattices, and partitions (2005).

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Dear @Carlo Beenakker Thanks for answer. –  veg_nw Mar 17 at 15:20

You want to have a simpler summation in Faa Di Bruno Formula by avoiding using specific Diophantine equation? See this referrences

Voinov, V. G., & Nikulin, M. (1994). On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications. Kybernetika, 30(3), 343-358.

Mortini, R. (2013). The Faa di Bruno formula revisited. Elemente der Mathematik, 68(1), 33-38.

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Dear @user48365 Thanks. Yes exactly I would like to have a simpler summation. –  veg_nw Mar 17 at 15:39

An algorithm in given in S. Blinnikov and R. Moessner, Expansions for nearly Gaussian distributions, Astron. Astrophys. Suppl. Ser. 130, 193 (1998).

(I'm not qualified to judge whether it's the best one or not)

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