A New Formula of q-Fubini Numbers via Goncharov polynomials

Connected the generalized Goncharov polynomials associated to a pair ($\mathrm{\partial },\mathcal{Z}$) if a delta operator $\mathrm{\partial }$ and an interpolation grid $\mathcal{Z}$, introduced by Lorentz, Tringali and Yan in [7], with the theory of binomial enumeration and order statistics, a new $q$-deformed of these polynomials given in this paper allows us to derive a new combinatorial formula of $q$-Fubini numbers. A combinatorial proof and some nice algebraic and analytic properties have been expanded to the $q$-deformed version.