Determinants, Choices and Combinatorics

TL;DR

This paper introduces a generalized formula that unifies previous results related to combinatorial determinants and applies it to prove new results connected to Rota's basis conjecture and the Atiyah-Sutcliffe problem.

Contribution

It presents a new formula that generalizes existing combinatorial determinant formulas and demonstrates its application to problems in combinatorics and geometry.

Findings

Unified formula encompassing Onn's and Svrtan's results

Proved new results related to Rota's basis conjecture

Extended Svrtan's argument to a combinatorial setting

Abstract

We prove a formula which generalizes both Onn's colorful determinantal formula, related to Rota's basis conjecture, and Svrtan's $n!$ formula, related to the Atiyah-Sutcliffe problem. In some cases, our formula allows us to prove some results similar in spirit to the statement of Rota's basis conjecture. We prove such a result using Svrtan's $n!$ formula, generalizing one of Svrtan's arguments to a combinatorial setting.