The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements

TL;DR

This paper investigates resistances in hypercubes, deriving exact formulas, generating functions, and asymptotic expansions, and explores their combinatorial interpretations to deepen understanding of electrical properties in high-dimensional structures.

Contribution

It introduces new exact expressions and asymptotic formulas for resistances in hypercubes, along with combinatorial insights into the expansion coefficients.

Findings

Exact resistance formulas for hypercube vertices

Asymptotic expansions of resistance values

Combinatorial interpretations of expansion coefficients

Abstract

Motivated by the problem of finding resistances among vertices in a hypercube, we derive exact expressions, generating functions, and asymptotic expansions for these resistances, then study the combinatorial interpretations of the coefficients arising in these asymptotic expansions.