I need to find the number of terms in a partial Bell polynomial, defined as $$B_{k,j}(x_{1}, x_{2}, \ldots, x_{k-j+1}) = \sum \frac{k!}{m_{1}! \ m_{2}! \ldots m_{k-j+1}!} \prod_{\ell=1}^{k-j+1} \bigg( \frac{x_{\ell}}{\ell!} \bigg)^{m_{\ell}}$$ where the sum is taken over all sequences $\{m_{\ell}\}_{\ell=1}^{k-j+1}$ of non-negative integers satisfying $$\sum_{i=1}^{k-j+1} m_{i} = j$$ $$\sum_{i=1}^{k-j+1} i m_{i} = k.$$ How can I find the number sequences $\{m_{\ell}\}_{\ell=1}^{k-j+1}$ satisfying the above equations? I have tried to solve them individually for several pairs $(k,j)$ but I'm not able to find a general formula.
How many terms in a partial Bell polynomial
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sequences-and-series
combinatorics
polynomials
summation
1 Answers
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Bell polynomials are partition polynomials. They count the number of partitions of sets. The sequences $\{m_l\}_{l=1}^{k-j+1}$ are the partitions of integer $k$ into $j$ parts: A008284.
There are no explicit non-recursive formulas for the Bell polynomials, for the number of terms in a partial Bell polynomial or for the partitions. You have to use software programs, generating functions or recurrence equations.