On Bergeron's positivity problem for q -binomial coefficients

Authors

Fabrizio Zanello, Michigan Technological UniversityFollow

Document Type

Article

Publication Date

4-27-2018

Abstract

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.

Publisher's Statement

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

Publication Title

The Electronic Journal of Combinatorics

Creative Commons License

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

Recommended Citation

Zanello, F. (2018). On Bergeron's positivity problem for q -binomial coefficients. The Electronic Journal of Combinatorics, 25(2), 1-9. http://doi.org/10.37236/7358
Retrieved from: https://digitalcommons.mtu.edu/math-fp/63