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<p>We conjecture that, if the quotient of two <i>q</i>-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D.&#xa0;Stanton’s fake Gaussian sequences. As a corollary of one of our results we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S.&#xa0;Hopkins and M.&#xa0;Rubey, <i>Selecta Math. (N.S.)</i> <b>28</b> (2022), Paper No.&#xa0;10] is indeed a polynomial with non-negative integer coefficients.</p>

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A positivity conjecture for a quotient of q-binomial coefficients

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  • 发表日期：

    2025年12月17日

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  • M. Gatzweiler,
  • C. Krattenthaler

摘要

We conjecture that, if the quotient of two q-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D. Stanton’s fake Gaussian sequences. As a corollary of one of our results we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S. Hopkins and M. Rubey, Selecta Math. (N.S.) 28 (2022), Paper No. 10] is indeed a polynomial with non-negative integer coefficients.