By Elaine Wong
Technical papers can be enjoyable to read too! The authors bring you on a journey into the binomial determinant world and invite you to sit next to them on their roller coaster ride as they tackle the proof of a particular determinant formula, one that can be related to various combinatorial applications. In the process, they use three well-known determinantal techniques (namely: Dodgson's condensation, constant term evaluations, holonomic ansatz) and skillfully show their application to the target problem. What is notable is not just the elegance and deftness with which the methods are adapted, but the ability of the methods themselves to harness different aspects of the underlying structure of determinants and binomial coefficients to be used for the `unrolling' and `reorganization' of its factors into a well-formed product formula. This is instructive to see side-by-side and in a notationally consistent manner. The authors then take a detour to establish a class of matrices with a similar underlying structure (i.e., sums and differences of Toeplitz and Hankel matrices) that would lend itself well to the methods of this paper, as well as a tangential combinatorial connection to the enumeration of plane partitions (applications never hurt, right?), before arriving at the climax of the story, where a fourth technique appears in the form of `this is just a specific case of a more general theorem in a very well-known determinant paper', and the disappointment at discovering this is palpable. But, the authors are all brilliant teachers and they turn their little exercise into a showcase of academic honesty with a splash of positive attitude, all for our pedagogical benefit.