A Proper Subclass of the Separable Permutations Whose Generating Function Lies Outside Q(z, √(1−4z), √(1−6z+5z²))

Question 5 of R. Brignall's abstract (joint work with M. B\'ona, C. Defant, M. Opler and V. Vatter) in Oberwolfach Report 6/2024, p. 292, asks whether every proper subclass of the separable permutations \(\operatorname{Av}(2413,3142)\) has a generating function in \(K=\mathbb{Q}\bigl(z,\sqrt{1-4z},\sqrt{1-6z+5z^2}\bigr)\). The answer is no. The class \(\mathcal{C}=\operatorname{Av}(2341,2413,3142)\) is a proper subclass of the separable permutations which contains \(\operatorname{Av}(231)\). Its generating function \(f=\sum_{n\ge0}|\mathcal{C}_n|z^n\) satisfies \(z(1-z+z^2)f^2-(1-2z+2z^2)f+(1-z)^2=0\), so that \(f=\bigl(1-2z+2z^2-\sqrt D\bigr)/\bigl(2z(1-z+z^2)\bigr)\) with \(D=1-8z+20z^2-24z^3+16z^4-4z^5\). This enumeration is due to Callan, Mansour and Shattuck (Discrete Math. Theor. Comput. Sci. 19:1 (2017), \#6, Theorem 11), and the same generating function is in the PermPAL database. The polynomial \(D\) is irreducible over \(\mathbb{Q}\) and coprime to \((1-4z)(1-z)(1-5z)\), whereas a nonzero element of \(\mathbb{Q}(z)\) is a square in \(K\) only if it lies in the square class of \(1\), \(1-4z\), \((1-z)(1-5z)\) or \((1-4z)(1-z)(1-5z)\); hence \(\sqrt D\notin K\) and \(f\notin K\). The contribution of this note is the observation that this class answers Question 5 negatively, a self-contained proof of its enumeration by the substitution decomposition, and an exact certificate that its generating function is not in \(K\). This is an unrefereed note.