Determinants, Choices and Combinatorics

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.