Stratified motivic invariants and bivariate deformations of Poincaré polynomials

For a stratified variety $X$ and a motivic invariant $\mathbb{H}$, we consider the stratified invariant which interpolates the invariant of $X$ and that of its interior $X^{\circ}$. Based on observations in moduli theory, we introduce the notion of an echelon tower of stratified varieties and then prove explicit inductive formulae for the stratified invariants of the moduli spaces $\overline{M}_{0,n}$ of stable curves of genus $0$ and the Fulton-MacPherson varieties $Y[n]$ for any smooth projective variety $Y$. Using these, we show that the mysterious bivariate deformations of the even degree Poincar\'e polynomials of $\overline{M}_{0,n+1}$ and $\mathbb{P}^1[n]$ in \cite{BercziKiem2026} are nothing but stratified virtual Poincar\'e polynomials.