First Written: June 22nd, 2017; This version: April 16th, 2018.

Abstract: The q-binomial coefficients were assumed to be unimodal as early as the 1850's, but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an ''elementary'' proof using linear algebra. Finally, in 1989, Kathy O'Hara provided a combinatorial proof of the unimodality of the q-binomial coefficients. Very soon thereafter, Doron Zeilberger translated the argument into an elegant recurrence. We introduce several perturbations to the recurrence to create a larger family of unimodal polynomials. We analyze how these perturbations affect the final polynomial and analyze some specific cases.

Maple packages

• Gnk.txt, a Maple package for discovery of symmetric and unimodal polynomials.

Sample Input and Output for Gnk.txt

If you want lists of rational functions in q^n that are guaranteed to be symmetric and unimodal polynomials for integer n, you can try fixing k and restricting to distinct (important) partitions as in the following:

• Distinct Partitions

• Distance 1, Minimum part 2

• Distance 1, Minimum part 3

• Distance 1, Minimum part 4

• Distance 2, Minimum part 1

• Distance 2, Minimum part 2

• Distance 2, Minimum part 3

• Distance 2, Minimum part 4

• Distance 3, Minimum part 1

• Distance 3, Minimum part 2

• Distance 3, Minimum part 3

• Distance 3, Minimum part 4

• Distance 4, Minimum part 1

• Distance 4, Minimum part 2

• Distance 4, Minimum part 3

• Distance 4, Minimum part 4

  These polynomials are only guaranteed to be symmetric and unimodal for n>=2*k-2 (because of recurrence calls). The "Combined" versions are very surprisingly guaranteed to be symmetric and unimodal. To see the proof, look at the "Decomposed" version and use observations about darga to complete the reasoning.

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