Kathy O'Hara's Constructive Proof of the Unimodality of the Gaussian Polynomials

By Doron Zeilberger

Appeared in: Amer. Math. Monthly 96 (1989), 590-602.

Ever since I was a (mathematical) child, I dreamed of finding a constructive proof of the unimodality of the Gaussina polynomials. The same can be said about almost every bijective combinatorialist. Hence Kathy O' Hara's tour-de-force first met with considerable skepticism, especially since it was rather complicated.

I invested lots of time and energy trying to decipher O'Hara's complicated YET beautiful construction. I was rewarded amply. First, it was great fun to understand. Second, I realized that her construction implies an amazing (not-shaloshable) q-binomial coefficient identity that implied a conjecture of Andrew Odlyzko (see my paper with D. Stanton: Proc. Amer. Math. Soc. 107 (1989), 39-42), which in turn lead to further research by Jane Friedman, Stanton and Joichi, and most recently, George Andrews, among others. Third, I got to write this expository paper, that won me the prestigious Lester Ford award for the "best paper of the year" of the Amer. Math. Monthly. I got it in the summer of 1990, right before I turned forty, and it was a consolation prize to the terrible realization that I will never be a Fields medalist.

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Added Dec. 7, 2022: Read Xiangdong Wen's Amazing SCD for L(5,n)

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