A Terrible Expansion of the Determinant

TL;DR

This paper introduces a new, more extensive expansion formula for the determinant derived from multivariate finite operator calculus, utilizing combinatorial techniques and properties of the permutahedron.

Contribution

It presents a novel determinant expansion formula with more terms than traditional methods, based on transfer formulas and combinatorial structures.

Findings

Derived a new determinant expansion formula

Utilized properties of the permutahedron in proof

Compared the new expansion to Ryser's formula for the permanent

Abstract

From a transfer formula in multivariate finite operator calculus, comes an expansion for the determinant similar to Ryser's formula for the permanent. Although this one contains many more terms than the usual determinant formula. To prove it, we consider the poset of ordered partitions, properties of the permutahedron, and some good old fashioned combinatorial techniques.