bibkey: oh2018identities authors: Se-jin Oh, Travis Scrimshaw year: 2018 title: “Identities from representation theory” doi: 10.48550/arXiv.1805.00113 url: https://arxiv.org/abs/1805.00113v1 claim: “The paper derives determinant and Pfaffian identities from representation theory and, in an appendix on other q-determinants, conjectures on numerical evidence that the two-shifted Hankel determinants of Cigler’s q-Motzkin numbers factor as q^(c_n) f_n(q) for an explicit f_n.” strata_touched:

• D5/S0/Certificates/OhScrimshawQMotzkinHankelRefutation license: citation-only triage: anchor

Identities from representation theory

The appendix “Other q-determinants” of Oh and Scrimshaw considers the q-Motzkin numbers defined by Cigler,

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,

recalls Cigler’s evaluations of the unshifted and one-shifted Hankel determinants, and states, “based on numerical computations”, the conjecture labelled conj:factored_motzkin_2shifted:

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}.

The same label is used again in the source file for a second conjecture on the three-shifted determinants, followed by a positivity conjecture conj:factored_motzkin_positive for all even shifts. At q = 1 the numbers M̃†_n(1) are the Motzkin numbers 1, 1, 2, 4, 9, 21, 51, …, since both satisfy the first-return recursion.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1805.00113
• URL: https://arxiv.org/abs/1805.00113v1
• Version and location: arXiv:1805.00113v1 (2018-04-30), source file q_analogs_repr_theory.tex, section “Other $q$-determinants” of the appendix: the definition of Cigler’s q-Motzkin numbers and the first conjecture labelled conj:factored_motzkin_2shifted. The journal version, Discrete Math. 342(9) (2019) 2493–2541, DOI 10.1016/j.disc.2019.05.020, was not read.