# Combinatorial applications of the special numbers and polynomials

## TL;DR

This paper explores properties, combinatorial interpretations, and computation formulas of special numbers and polynomials, including Euler, Stirling, and factorial numbers, using generating functions and their relations to rook polynomials.

## Contribution

It introduces new properties and interpretations of special numbers and polynomials, connecting them with combinatorial structures and providing formulas for their computation.

## Key findings

- Derived new properties of special numbers and polynomials.
- Established combinatorial interpretations related to rook polynomials.
- Provided formulas for efficient computation of these numbers.

## Abstract

In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials, which are the Euler numbers, the Stirling numbers of the second kind, the central factorial numbers and the array polynomials. We also discuss some combinatorial interpretations of these numbers related to the rook polynomials and numbers. Furthermore, we give computation formulas for these numbers and polynomials.