Generating-Function Field for Separable Permutation Subclasses

Definitions and mathematical statement

For each integer $n\geq 0$, let $S_n$ be the set of permutations of $\{1,\ldots,n\}$, with $S_0$ consisting of the empty permutation. A permutation $\sigma\in S_k$ is a pattern of $\pi\in S_n$ if there exist indices $1\leq i_1<\cdots<i_k\leq n$ such that $\pi(i_a)<\pi(i_b)$ if and only if $\sigma(a)<\sigma(b)$ for all $a,b$. A permutation avoids $\sigma$ if it does not contain this pattern. For a set $B$ of permutations, write $\operatorname{Av}(B)$ for the set of finite permutations avoiding every member of $B$.

A permutation class is a set of finite permutations closed under taking patterns. Define the class of separable permutations by \[ \operatorname{Sep}=\operatorname{Av}(2413,3142). \] A proper subclass means a permutation class $\mathcal C\subsetneq\operatorname{Sep}$. Its ordinary generating function, counted by permutation length, is \[ F_{\mathcal C}(z)=\sum_{n\geq0}|\mathcal C\cap S_n|z^n\in{\mathbb{Q}}[[z]]. \] Let $u,v\in{\mathbb{Q}}[[z]]$ be the unique series satisfying \[ u^2=1-4z,\qquad v^2=1-6z+5z^2,\qquad u(0)=v(0)=1, \] and regard $K={\mathbb{Q}}(z,u,v)$ as a subfield of the formal Laurent series field ${\mathbb{Q}}((z))$.

Question. Is it true that \[ \forall\,\mathcal C\subsetneq\operatorname{Sep}\text{ that is a permutation class},\qquad F_{\mathcal C}(z)\in{\mathbb{Q}}\!\left(z,\sqrt{1-4z},\sqrt{1-6z+5z^2}\right)=K? \] Equivalently, does every such generating function admit a rational expression in $z,u,v$ with rational coefficients?

The supplied excerpt gives the contextual example \[ \mathcal D=\operatorname{Av}(2143,2413,3142),\qquad F_{\mathcal D}(z)=\frac{2}{1+z+\sqrt{1-6z+5z^2}}, \] and identifies its enumeration with OEIS sequence A033321. The supplied dated assessment describes this example as refuting the smaller field ${\mathbb{Q}}(z,\sqrt{1-4z})$, while leaving the displayed two-radical question unresolved. These are supplied source claims, not independently verified conclusions of this draft. The excerpt's suggested recursive explanation for the example is heuristic; its assertion of a smallest class lacks explicit initial conditions and is not an additional hypothesis of the question. The empty-permutation convention above is explicit; changing only the constant term does not affect membership in $K$.

Short English statement

Does every pattern-closed proper subclass of the separable permutations have its ordinary length-generating function in ${\mathbb{Q}}(z,\sqrt{1-4z},\sqrt{1-6z+5z^2})$? The supplied example motivates the second radical. The record labels the enlarged question open; this statement-only draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30005767$, title "Generating-Function Field for Separable Permutation Subclasses," exact source problem code catalogue entry. The record places the problem in the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Permutation Patterns (2024), Oberwolfach Reports, pp. 273--308, DOI 10.4171/owr/2024/6. Source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2024/6.
• Supplied literature-source link labelled "explicit generating-function paper": https://arxiv.org/abs/1510.06328. The supplied excerpt refers to a paper by three workshop participants as reference [5], but does not supply its bibliographic entry or explicitly establish its identification with this link.
• The excerpt identifies the example's enumeration as OEIS sequence A033321; no OEIS URL is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the question open and says that a concrete proper subclass refutes the smaller one-radical field while the enlarged two-radical question remains unresolved. This is supplied dated curation, not independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports a direct check of the official report. This is supplied verification metadata, not a source visit or verification performed for this draft.