A New Formula of q-Fubini Numbers via Goncharov polynomials

TL;DR

This paper introduces a new $q$-deformed formula for Fubini numbers by connecting Goncharov polynomials with binomial enumeration and order statistics, providing combinatorial, algebraic, and analytic insights.

Contribution

It presents a novel $q$-deformation of Goncharov polynomials linked to delta operators and interpolation grids, leading to a new combinatorial formula for $q$-Fubini numbers.

Findings

Derived a new combinatorial formula for $q$-Fubini numbers.

Provided combinatorial, algebraic, and analytic properties of the $q$-deformed polynomials.

Established connections between Goncharov polynomials, binomial enumeration, and order statistics.

Abstract

Connected the generalized Goncharov polynomials associated to a pair ($\partial,\mathcal{Z}$) if a delta operator $\partial$ 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.