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In search of a combinatorial reasoning for a vanishing sum
Assume $s, j \in\mathbb{N}$. Define the set
$$\mathcal{A}_{j,s}:=\{(n_1,n_2,\dots,n_j)\in\mathbb{Z}_{\geq0}^j\vert \,
n_1+2n_2+\cdots+jn_j=j, \, n_1+n_2+\cdots+n_j=s\}.$$
Question. Is there a ...
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