Schur-positivity and log-concavity of anti-Markov polynomials

Prove that, for every anti-Markov number $a\geq 2$, the anti-Markov polynomial $\widetilde{\{a\}}$ is Schur-positive and has a log-concave $(q,t)$)-expansion.

Background

The paper introduces symmetric (q,t)(q,t))-analogues of Markov numbers through a Markov-type equation and defines anti-Markov polynomials by applying the involution that changes the sign of e2e_2.

The authors state that computed examples suggest γ\gamma-anti-positivity for polynomials associated with odd-indexed Pell numbers. They establish Schur-positivity for the ordinary Markov polynomials in one proposition, report that log-concavity was checked computationally through a≤135137a\leq135137, and then formulate the unresolved conjecture for anti-Markov polynomials.

References

This pushes us to make the following conjecture (as we also checked it up to the same bound): \begin{conjecture} For all anti-Markov number $a\geq 2$, the polynomial $#1{a}$ is Schur-positive, and its $(q,t)$-expansion is log-concave. \end{conjecture}

— A $(q,t)$-Overview of $q$-Analogs  (2608.30979 - Bergeron, 31 Aug 2026) in Section 9, subsection “Markov numbers (polynomials),” immediately following the displayed computational evidence and before the subsequent examples