By Tewodros Amdeberhan, Christoph Koutschan, and Doron Zeilberger
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
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 Feb. 4, 2024: Read Elaine Wong's insightful MathSciNet review
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