We consider the family $\mathrm{MC}_d$ of monic centered polynomials of one complex variable with degree $d \geq 2$, and study the map $\widehatΦ_d:\mathrm{MC}_d\to \widetildeΛ_d \subset \mathbb{C}^d / \mathfrak{S}_d$ which maps each $f \in \mathrm{MC}_d$ to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber $\widehatΦ_d^{-1}\left(\barλ\right)$ for every $\barλ \in \widetildeΛ_d$ except when the fiber $\widehatΦ_d^{-1}\left(\barλ\right)$ contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama, The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132--185] which gave a rather long algorithm with some induction processes.