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We consider the family $\mathrm{MC}_d$ of monic centered polynomials of onecomplex variable with degree $d \geq 2$, and study the map$\widehat{\Phi}_d:\mathrm{MC}_d\to \widetilde{\Lambda}_d \subset \mathbb{C}^d /\mathfrak{S}_d$ which maps each $f \in \mathrm{MC}_d$ to its unorderedcollection of fixed-point multipliers. We give an explicit formula for countingthe number of elements of each fiber$\widehat{\Phi}_d^{-1}\left(\bar{\lambda}\right)$ for every $\bar{\lambda} \in\widetilde{\Lambda}_d$ except when the fiber$\widehat{\Phi}_d^{-1}\left(\bar{\lambda}\right)$ contains polynomials havingmultiple fixed points. This formula is not a recursive one, and is a drasticimprovement of our previous result [T. Sugiyama, The moduli space of polynomialmaps and their fixed-point multipliers. Adv. Math. 322 (2017), 132--185] whichgave a rather long algorithm with some induction processes.

The Moduli Space of Polynomial Maps and Their Fixed-Point Multipliers: II. Improvement to the Algorithm and Monic Centered Polynomials