Oh and Scrimshaw’s two-shifted q-Motzkin conjecture is false as printed

Abstract

At n = 3 the two-shifted Hankel determinant of Cigler’s q-Motzkin numbers takes the value 3 at q = 1, while the printed factor f_3 takes the value 2, so no power of q times f_3 equals the determinant.

Definition 1.1 (Cigler’s q-Motzkin numbers).

$$(\mathrm{qMotzkin}\left(q,0\right)=1)\wedge (\forall n\in \mathbb{N},\mathrm{qMotzkin}\left(q,n+1\right)=\mathrm{qMotzkin}\left(q,n\right)+\sum _{k\in \mathrm{Fin}\left(n\right)}{q}^{k+1}\cdot \mathrm{qMotzkin}\left(q,k\right)\cdot \mathrm{qMotzkin}\left(q,n-k-1\right))$$

Formalization. D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.qMotzkin (✓ std3).

Citation. Se-jin Oh, Travis Scrimshaw (2018). Identities from representation theory. DOI: 10.48550/arXiv.1805.00113. URL: https://arxiv.org/abs/1805.00113v1.

Commentary.

The recursion of the source over any commutative ring with parameter q: M(0) = 1 and M(n + 1) = M(n) + sum over k < n of q^(k+1) M(k) M(n - k - 1). The conjecture uses the polynomial ring Z[q] with q the variable.

Definition 1.2 (The printed factor).

$$\forall n\in \mathbb{N},\mathrm{fPrinted}\left(q,n\right)=\mathrm{ite}\left(\mathrm{NatMod}\left(n,3\right)=0,\sum _{k\in \{k\in Finset.Icc\left(1,n\right)\mid \mathrm{NatMod}\left(k,3\right)\ne 1\}}{q}^k,(q+1)\cdot \sum _{k\in Finset.range\left(\mathrm{NatDiv}\left(n,3\right)+1\right)}{q}^{3\cdot k}\right)$$

Formalization. D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.fPrinted (✓ std3).

Citation. Se-jin Oh, Travis Scrimshaw (2018). Identities from representation theory. DOI: 10.48550/arXiv.1805.00113. URL: https://arxiv.org/abs/1805.00113v1.

Commentary.

For n divisible by three the source sums q^k over 1 <= k <= n with k not congruent to 1 modulo 3; otherwise it multiplies q + 1 by the sum of q^(3k) over k <= floor(n/3). NatMod and NatDiv are the natural remainder and quotient.

Definition 1.3 (The two-shifted Hankel determinant).

$$\forall n\in \mathbb{N},\mathrm{hankelTwoShifted}\left(q,n\right)=\det \left([\mathrm{qMotzkin}\left(q,i+j+2\right){]}_{i,j\in \mathrm{Fin}\left(n\right)}\right)$$

Formalization. D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.hankelTwoShifted (✓ std3).

Citation. Se-jin Oh, Travis Scrimshaw (2018). Identities from representation theory. DOI: 10.48550/arXiv.1805.00113. URL: https://arxiv.org/abs/1805.00113v1.

Commentary.

The determinant of the n x n matrix with entries M(i + j + 2) for 0 <= i, j < n.

Definition 1.4 (The printed conjecture).

$$claim\iff (\forall n\in \mathbb{N},(1\le n)\Rightarrow (\exists c\in \mathbb{N},\mathrm{hankelTwoShifted}\left(X,n\right)=X^c\cdot \mathrm{fPrinted}\left(X,n\right)))$$

Formalization. D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.claim (✓ std3).

Citation. Se-jin Oh, Travis Scrimshaw (2018). Identities from representation theory. DOI: 10.48550/arXiv.1805.00113. URL: https://arxiv.org/abs/1805.00113v1.

Commentary.

For every n >= 1 some natural power of q times f_n equals the determinant in Z[q]. Only the first of the two conjectures that share the label conj:factored_motzkin_2shifted is stated.

Theorem 1.5 (The counterexample n = 3).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.result (✓ std3). ∎

Resolves. Problems/oh-scrimshaw-2018-two-shifted-q-motzkin-hankel-refutation (refuted) by D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.result.

Source. Repository-derived.

Acknowledgement. Se-jin Oh, Travis Scrimshaw (2018). Identities from representation theory. DOI: 10.48550/arXiv.1805.00113. URL: https://arxiv.org/abs/1805.00113v1.

Commentary.

Evaluation at q = 1 is a ring homomorphism from Z[q] to Z, so it commutes with the determinant and with the recursion. At q = 1 the recursion gives the Motzkin numbers 1, 1, 2, 4, 9, 21, 51, and the determinant becomes det[[2, 4, 9], [4, 9, 21], [9, 21, 51]] = 3. The printed side becomes 1^c f_3(1), and f_3 = q^2 + q^3 gives 2. So the identity fails at n = 3 for every c.

References

• Truth anchor: D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.claim
• Truth anchor: D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.fPrinted
• Truth anchor: D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.hankelTwoShifted
• Truth anchor: D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.qMotzkin
• Truth anchor: D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation.result