Remarks on the inverse problem of the collinear central configurations in the $N$-body problem

For a fixed configuration in the collinear $N$-body problem, the existence of the central configurations is determined by a system of linear equations, which in turn is determined by certain Pfaffians for even or odd $N$ in literatures. In this short note, we prove that the Pfaffians of the associate matrices for all even number collinear configurations are nonzero if and only if the extended Pfaffians of the associate matrices for all odd number collinear configurations are nonzero. Therefore, the inverse problem of the collinear central configurations can be answered and each collinear configuration determines a one-parameter family of masses with a fixed total mass if the Pfaffians of the associate matrix for all collinear even number bodies are nonzero. We also make some remarks on the super central configurations and the number of collinear central configurations under different equivalences, especially a lower bound for the number of collinear central configurations under the geometric equivalence.