slug: oh-scrimshaw-2018-two-shifted-q-motzkin-hankel-refutation bibkey: oh2018identities doi: 10.48550/arXiv.1805.00113 url: https://arxiv.org/abs/1805.00113v1 triage: theorem motivation_gids:
- D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.result
Oh and Scrimshaw’s two-shifted q-Motzkin Hankel conjecture
Problem
Oh and Scrimshaw consider Cigler’s q-Motzkin numbers
M̃†_{n+1}(q) = M̃†n(q) + Σ{k=0}^{n−1} q^{k+1} M̃†k(q) M̃†{n−k−1}(q), M̃†_0(q) = 1,
and state, based on numerical computations, the first conjecture labelled
conj:factored_motzkin_2shifted in their appendix on other q-determinants:
Define f_n(q) := Σ_{1 ≤ k ≤ n, k ≢ 1 mod 3} q^k if n ≡ 0 mod 3, and (q+1)(Σ_{k=0}^{⌊n/3⌋} q^{3k}) otherwise. Then we have (det[M̃†{i+j+2}(q)]{i,j=0}^{n−1}){n=1}^∞ = (q^{c_n} f_n(q)){n=1}^∞ for some c_n ∈ ℤ_{≥0}.
Issue #10046 fixes the readings: the polynomials and the determinant live in
ℤ[q]; “for some c_n” means that for each n ≥ 1 some natural number
c_n makes the polynomial identity hold; the second conjecture carrying the
same label, on the three-shifted determinants, is not addressed.
Motivation
The frozen declaration
D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.result proves that
the conjecture is false as printed: at n = 3 no power of q times f_3
equals the determinant. The formal statement makes no claim about a corrected
summation range.
Gap
Issue #10046 preregisters the published conjecture and its literature check.
MathDB /p/339076 has status open with zero solutions. Semantic Scholar
lists four citing papers, on fluctuations of Young diagrams for symplectic
groups, branching formulae for classical groups, promotion for fans of Dyck
paths, and skew Howe duality; none concerns q-Motzkin Hankel determinants.
These readings are not-found-in-searched-scope; they do not establish an
exhaustive worldwide literature search, priority, or the absence of an
independent refutation.
Route
Evaluation at q = 1 is a ring homomorphism ℤ[q] → ℤ, so it commutes with
the determinant and, by strong induction, with the recursion. At q = 1 the
recursion is the first-return recursion of the Motzkin numbers, giving
1, 1, 2, 4, 9, 21, 51, and the determinant becomes
det[[2, 4, 9], [4, 9, 21], [9, 21, 51]] = 36 − 60 + 27 = 3.
On the printed side f_3(q) = q^2 + q^3, so 1^c f_3(1) = 2 for every c.
Falsifier
The refutation would fail if some reading of the printed f_3 evaluated to
3 at q = 1; the printed range 1 ≤ k ≤ 3 with k ≢ 1 (mod 3) contains
only k = 2, 3.
Evidence
Exact computation in ℤ[q], with the determinant expanded over column
subsets, gives det = q^{11}(1 + q^2 + q^3) at n = 3. For n = 1, …, 12
the printed statement holds for n ≢ 0 (mod 3) and fails for
n = 3, 6, 9, 12, where the determinant at q = 1 is 3, 5, 7, 9 against
f_n(1) = 2, 4, 6, 8. Extending the first range to 0 ≤ k ≤ n makes every
case n ≤ 12 hold, with c_n = 0, 4, 11, 26, 51, 85, 133, 197, 276, 375, 496, 638. In Lean, the same definitions give the determinant 2 at q = 1
for n = 1, 2 and f_1(1) = f_2(1) = 2, where the printed statement holds.
The Lean proof has only the standard axiom closure propext,
Classical.choice, and Quot.sound.
Triage
Tier 1 external named conjecture, preregistered in issue #10046 before the
probe. The result has proof_shape: bind-only, escape_witness: null, and
admission_basis: open-problem-resolution. Its computational use is a
certified-instance with basis=refutes: the result negates the closed
claim. Resolution: refuted, as printed. The statement with the corrected
range, the three-shifted conjecture and the positivity conjecture are outside
its scope.
ASSUMED-UNVERIFIED
Whether the journal version, Discrete Math. 342(9) (2019) 2493–2541, keeps
the appendix and the printed range was not checked. The corrected range is
supported only for n ≤ 12 and is not claimed. The bounded literature check
does not establish exhaustive worldwide novelty, priority, or the absence of
an independent refutation. The Lean kernel does not authenticate the external
source or its version history.