Refinements of the Bell and Stirling numbers

TL;DR

This paper introduces new unified refinements of classical combinatorial numbers, deriving recurrence relations and closed-form formulas by combining analytic combinatorics, umbral calculus, and probability theory.

Contribution

It presents novel refinements of Bell and Stirling numbers that unify various related combinatorial sequences and provides new recurrence relations and explicit formulas.

Findings

Unified framework for Bell, Stirling, and related numbers

Derived new recurrence relations for refined numbers

Obtained explicit sum formulas over compositions

Abstract

We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial, restricted Bell, and $r$-derangement numbers (and probably more!). By combining methods from analytic combinatorics, umbral calculus, and probability theory, we derive several recurrence relations and closed form expressions for these numbers. By specializing our results to the classical case, we recover explicit formulae for the Bell and Stirling numbers as sums over compositions.