The Telephone Exchange Problem Revisited: A Combinatorial Approach

TL;DR

This paper revisits the telephone exchange problem by exploring generalized Bessel polynomials, analyzing their combinatorial properties, and establishing connections with Whitney and Dowling numbers to deepen understanding of the problem's mathematical structure.

Contribution

It introduces two generalizations of Bessel polynomials and links their combinatorial properties to Whitney and Dowling numbers, offering new insights into the problem.

Findings

Established relationships between generalized Bessel polynomials and Whitney numbers.

Connected Dowling numbers to the combinatorial properties of the polynomials.

Provided a new combinatorial framework for analyzing the telephone exchange problem.

Abstract

In this study we revisit the telephone exchange problem. We discuss a generalization of the telephone exchange problem by discuss two generalizations of the Bessel polynomials. We study combinatorial properties of these polynomials, and show how the numbers are related to the well known Whitney numbers and Dowling numbers