By Tewodros Amdeberhan, Christoph Koutschan, and Doron Zeilberger

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First Written: July 4, 2023. This version: July 9, 2023.

A revised version [following suggestions of the "birthday boy"] Appeared, with another title (also suggested by Christian K.), in Séminaire Lotharingien de Combinatoire, B89a (2023), 14 pp

Dedicated to Maestro Christian Krattenthaler (b. 8 Oct., 1958) on his millionth-and-one birthday

In this case study, we hope to show why Sheldon Axler was not just wrong, but
really 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.

Added July 5, 2023: Remark from Christian Krattenthaler on the first version  :

Concerning the determinant that you look at: it is the special case of (3.18) in my "Advanced Determinant Calculus", where q=1, A=x+m, L_i=i-m. This identity *has* a simple direct proof. As explained above Theorem 30 in ADC, after having taken out appropriate factors, the determinant turns out to be a special case of Lemma 5 in ADC. That lemma is so general so that it can be proved in various ways, onoe of which is explained in Ref. [88] of ADC.

Added July 8, 2023: This has been incorporated in the current, much improved version.

Added Feb. 4, 2024: Read Elaine Wong's insightful MathSciNet review  

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