A Fibonacci sequence for linear structures with two types of components

TL;DR

This paper counts hierarchical binary voting systems with two voter types, deriving a closed-form Fibonacci-based formula for their enumeration as the number of voters varies.

Contribution

It introduces a novel counting method for hierarchical two-type voter systems, resulting in a Fibonacci sequence formula for the number of such systems.

Findings

Derived a closed-form Fibonacci sequence formula

Counted voting systems up to isomorphism

Established polynomial variation with voter number

Abstract

We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to count, up to isomorphism, the number of these voting systems for an arbitrary number of voters. We obtain a closed formula for the number of these systems, this formula follows a Fibonacci sequence with a smooth polynomial variation on the number of voters.