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bibkey: petrovscott2026clone
authors: Leonid Petrov and Jeanne Scott
year: 2026
title: Random Fibonacci words via clone Schur functions
doi: 10.1017/fms.2025.10165
url: https://arxiv.org/abs/2412.21126v2
claim: "Conjecture 6.3: There exist values of ρ and q for which the following specialization (which we call the alternative q-Charlier) is Fibonacci positive."
strata_touched:
  - D5/S3/Combinatorics/YoungFibonacci/PetrovScottAltQCharlierPositivity
license: citation-only
triage: anchor

Verified locator

DOI: 10.1017/fms.2025.10165

Source: https://arxiv.org/abs/2412.21126v2

Forum of Mathematics, Sigma 14 (2026), e15. The locators below refer to the pagination of arXiv version 2.

Source statements

Define two sequences of tridiagonal determinants as follows:

Equation (2.10), p. 15, defines the determinants of the two displayed tridiagonal matrices. The first has diagonal x₁, x₂, …, superdiagonal y₁, y₂, … and subdiagonal one. The second has first row y₁, x₁y₂, 0, …, then diagonal x₃, x₄, …, superdiagonal y₃, y₄, … and subdiagonal one.

Here ℓ ≥ 0. For a sequence u⃗ = (u₁, u₂, …), denote its shift by u⃗ + ℓ = (u₁₊ℓ, u₂₊ℓ, …), where ℓ ∈ ℤ≥0.

Definition 2.5, p. 15:

For any Fibonacci word w, define the (biserial) clone Schur function s_w(x⃗ | y⃗) through the following recurrence: s_w(x⃗ | y⃗) := A_k(x⃗ | y⃗), if w = 1^k for some k ≥ 0; B_k(x⃗ + |u| | y⃗ + |u|) · s_u(x⃗ | y⃗), if w = 1^k2u for some k ≥ 0.

Definition 4.1, p. 21:

If s_w(x⃗ | y⃗) > 0 for all w ∈ 𝕐𝔽, we say that (x⃗, y⃗) is Fibonacci positive.

Conjecture 6.3, p. 31:

There exist values of ρ and q for which the following specialization (which we call the alternative q-Charlier) is Fibonacci positive:

$$x_k=\frac{q^{k-1}(1+\rho q^{k-2}+\rho q^{k-1}-\rho q^{2k-2})}{(1+\rho q^{2k-3})(1+\rho q^{2k-1})},y_k=\frac{\rho q^{3k-2}(1-q^k)(1+\rho q^{k-1})}{(1+\rho q^{2k})(1+\rho q^{2k-1}{)}^2(1+\rho q^{2k-2})},k\ge 1.$$

Mathematical correspondence

Words are lists over {1, 2}; their weight is the sum of their letters. Sequences are functions from Nat to Real, used from index one. The source’s shift is written as fun k => u (k + offset) inside the recursion. The auxiliary recursion counts leading ones; s starts that count at zero. xAQC and yAQC retain the displayed rational expressions and integer exponents. The existential proposition uses these two sequences without restricting either real parameter; the proof takes both equal to 1/4.

Literature boundary

The source and the literature screen in 18756 identify Conjecture 6.3 as an open positivity problem. Completeness of that prior-art screen is ASSUMED-UNVERIFIED. The Lean result establishes the existential assertion, without an exact positivity region or a scaling limit.