Yay for Determinants!

TL;DR

This paper defends the importance of determinants in combinatorics, illustrating their utility through multiple methods to prove a complex determinantal formula, countering the dismissive stance of Sheldon Axler.

Contribution

The paper demonstrates the effectiveness of various determinant evaluation techniques in combinatorics, providing a detailed case study and introducing a new, efficient method for determinant calculation.

Findings

Determinants are valuable in enumerative combinatorics.

Multiple methods can be combined to evaluate complex determinants.

A new efficient determinant evaluation method is discussed.

Abstract

In this {\it case study}, we hope to show why Sheldon Axler was not just wrong, but {\em wrong}, when he urged, in 1995: ``Down with Determinants!''. We first recall how determinants are useful in enumerative combinatorics, and then illustrate three versatile tools (Dodgson's condensation, the holonomic ansatz and constant term evaluations) to operate in tandem to prove a certain intriguing determinantal formula conjectured by the first author. We conclude with a postscript describing yet another, much more efficient, method for evaluating determinants: `ask determinant-guru, Christian Krattenthaler', but advise people only to use it as a last resort, since if we would have used this last method right away, we would not have had the fun of doing it all by ourselves.