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The 1/2-conjecture for q-binomial coefficients with fractional index

For r∈Q+r\in\mathbb{Q}^+ and an integer k≥1k\geq 1, define the integer trace of the generalized qq-binomial coefficient by retaining its integer exponents, and write

[r+kk]q∣Z=∑d≥0cr(d)qd.\left[ \genfrac{}{}{0pt}{}{r+k}{k} \right]_q\big|_{\mathbb{Z}}=\sum_{d\geq 0}c_r(d)q^d.

For power series with nonnegative coefficients, write A(q)≥qB(q)A(q)\geq_q B(q) when every coefficient of A(q)−B(q)A(q)-B(q) is nonnegative. The 1/21/2-conjecture. For every r∈Q+r\in\mathbb{Q}^+ and every integer k≥1k\geq 1,

[1/2+kk]q≥q[r+kk]q.\left[ \genfrac{}{}{0pt}{}{1/2+k}{k} \right]_q\geq_q\left[ \genfrac{}{}{0pt}{}{r+k}{k} \right]_q.

Thus r=12r=\tfrac12 would be a global maximizer of the integer-trace coefficients in the coefficientwise partial order. The conjecture is motivated by the nonnegative signs and maximal integer support arising for r=12r=\tfrac12; the paper reports computational verification for r=13r=\tfrac13 and r=14r=\tfrac14 when 1≤k≤1501\leq k\leq150, but no general proof is given.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The 1/2-conjecture for q-binomial coefficients with fractional index

    Let k≥1k\geq 1 be an integer, and let [ak]q\left[\genfrac{}{}{0pt}{}{a}{k}\right]_q denote the fractional-index qq-binomial coefficient. The relation ≥q\geq_q is coefficientwise comparison of the associated integer trace expansions.

    The 1/2-conjecture. For all k≥1k\geq 1,

    [1/2+kk]q≥q[1/4+kk]q.\left[\genfrac{}{}{0pt}{}{1/2+k}{k}\right]_q \geq_q \left[\genfrac{}{}{0pt}{}{1/4+k}{k}\right]_q.

    This is the stated r=14r=\tfrac{1}{4} special case of the broader claim that r=12r=\tfrac{1}{2} globally maximizes the integer trace coefficients in the coefficientwise partial order. Its resolution is not specified in the supplied text.

    source: Guo-Niu Han and Huan Xiong, “The 1/2-Conjecture for q-Binomial Coefficients with Fractional Index”, arXiv:2606.01919 (2026).

Sources & referencesView supporting material

Primary source

Guo-Niu Han and Huan Xiong, “The 1/2-Conjecture for q-Binomial Coefficients with Fractional Index”, arXiv:2606.01919 (2026).

combinatoricsnumber-theoryq-binomial-coefficients

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