The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index
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2026
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| _version_ | 1866911739670429696 |
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| author | Han, Guo-Niu Xiong, Huan |
| author_facet | Han, Guo-Niu Xiong, Huan |
| contents | For a nonnegative integer $k$ and a rational number $r\in\mathbb{Q}^+$, we define the generalized Gaussian binomial coefficient $\qbinom{r+k}{k} = \frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\geq 2$, expanding $\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\qbinom{r+k}{k}$ is a \emph{Puiseux series} in $q$ with nonnegative exponents; concretely it lies in $\mathbb{Q}[[q^{1/b}]]$. The notion we single out is the \emph{integer trace} of this expansion, the subseries consisting of those terms $c_r(d)\,q^d$ whose exponent $d$ is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as $r$ varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \emph{$\tfrac{1}{2}$-Conjecture}: among all $r\in\mathbb{Q}^+$, the value $r=\tfrac{1}{2}$ maximizes the integer trace, in the sense that the coefficients of $\qbinom{1/2+k}{k}$ dominate those of $\qbinom{r+k}{k}$ coefficientwise for every $r$. That so elementary a definition should single out $\tfrac{1}{2}$ this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01919 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index Han, Guo-Niu Xiong, Huan Combinatorics Number Theory 05A20, 05A30 For a nonnegative integer $k$ and a rational number $r\in\mathbb{Q}^+$, we define the generalized Gaussian binomial coefficient $\qbinom{r+k}{k} = \frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\geq 2$, expanding $\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\qbinom{r+k}{k}$ is a \emph{Puiseux series} in $q$ with nonnegative exponents; concretely it lies in $\mathbb{Q}[[q^{1/b}]]$. The notion we single out is the \emph{integer trace} of this expansion, the subseries consisting of those terms $c_r(d)\,q^d$ whose exponent $d$ is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as $r$ varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \emph{$\tfrac{1}{2}$-Conjecture}: among all $r\in\mathbb{Q}^+$, the value $r=\tfrac{1}{2}$ maximizes the integer trace, in the sense that the coefficients of $\qbinom{1/2+k}{k}$ dominate those of $\qbinom{r+k}{k}$ coefficientwise for every $r$. That so elementary a definition should single out $\tfrac{1}{2}$ this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence. |
| title | The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index |
| topic | Combinatorics Number Theory 05A20, 05A30 |
| url | https://arxiv.org/abs/2606.01919 |