On Bergeron's positivity problem for q -binomial coefficients

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F. Bergeron recently asked the intriguing question whether ((b+c)/b)q-((a+d)/d)q has nonnegative coefficients as a polynomial in q, whenever a, b, c, dare positive integers, a is the smallest, and ad=bc. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for a≤3 and any b, c≥4. The main ingredient will be a novel (and rather technical) applicationof Zeilberger’s KOH theorem.

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© The author. Released under the CC BY-ND license (International 4.0). Publisher’s version of record: https://doi.org/10.37236/7358

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The Electronic Journal of Combinatorics

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Creative Commons Attribution-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-No Derivative Works 4.0 License.