Denote $(x)_t = x(x-1)(x-2)\cdots(x-t+1)$ and fix some $t_1,\dots,t_n\in\mathbb{N}$. Now consider the polynomials $$f_n(x)=\sum_{\pi\in L[n]}(-1)^{\vert\pi\vert-1}(\vert\pi\vert-1)!\prod_{A\in\pi}(x)_{\sigma(A)}$$ where the sum extends over all non-empty set-partitions $L[n]$ of $[n]:=\{1,\dots,n\}$ and $\sigma(A)=\sum_{i\in A}t_i$.

**Remark.** $\#L[n]=B_n$ the Bell numbers.

**Example.** Take $n=3$. Then the set of set-partitions of $[3]$ reads
$$L[3]=\{\{1,2,3\}, \{\{1,2\},\{3\}\}, \{\{1,3\},\{2\}\}, \{\{2,3\},\{1\}\}, \{\{1\},\{2\},\{3\}\}\}$$
and the polynomial becomes
$$f_3(x)=(x)_{t_1+t_2+t_3}-(x)_{t_1+t_2}(x)_{t_3}-(x)_{t_1+t_3}(x)_{t_2}-(x)_{t_2+t_3}(x)_{t_1}+2(x)_{t_1}(x)_{t_2}(x)_{t_3}.$$

Conjecture.The polynomial $f_n$ is of degree $1+(t_1-1)+\cdots+(t_n-1)$ in the variable $x$. Is this true?

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