Han–Xiong trace comparison for fractional Gaussian binomial coefficients

For positive rationals u=c/d and r=a/b in lowest terms with d even, d dividing lcm(b,2) and u≤r, does the integer trace of the fractional Gaussian binomial coefficient at u dominate that at r coefficient by coefficient for every k≥1? The Han–Xiong conjecture includes parameters below r=1/2.

Why it matters and what counts as progress

Why it matters: The comparison unifies fractional Gaussian-binomial trace inequalities across a broad rational parameter range while isolating the remaining small parameters.

What counts as progress: Prove the comparison on a larger parameter range or resolve the remaining canonical parameters r=1/(2m), with all denominator conditions retained.

How the problem developed

1 Jun 2026 · Conjecture [1]
Guo-Niu Han and Huan Xiong Defined the integer-exponent trace for fractional Gaussian binomial coefficients and conjectured coefficientwise maximality at r=1/2; proved several special cases.
31 Jul 2026 · Partial result [2]
Ken Ono Proved support dominance that settles the comparison for r≥1/2 and reduces the full conjecture to r=1/(2m); a computation checks all rational r through k=200.
24 Oct 2026 · Candidate proof [3]
Justin Du Posted a candidate all-r, all-k proof; the author says it has no external peer review or proof-assistant formalization and that generative AI contributed substantially.

AI contribution

On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients

Ken Ono

For positive rational u=c/d and r=a/b in lowest terms, with d even, d dividing lcm(b,2), and u≤r, the integer trace of the fractional Gaussian binomial coefficient at u dominates that at r coefficientwise for every k≥1. This proves the Han–Xiong comparison for every r≥1/2 and reduces the remaining parameters to 1/(2m).

AI involvementAI-autonomous

ImportanceNotable

Technical verificationFormal (Lean 4.31 with Mathlib and AxiomLib)

Human understandingFull