Wilf Class Asymptotics with Fixed Pattern Occurrences

Definitions and mathematical statement

Under the standard classical-pattern interpretation, let $S_n$ be the permutations of $\{1,\ldots,n\}$, with $S_0$ consisting of the empty permutation. For $\tau\in S_m$ and $\pi\in S_n$, define \[ \operatorname{occ}_{\tau}(\pi)=\#\{(i_1,\ldots,i_m):1\le i_1<\cdots<i_m\le n,\ \pi_{i_1}\cdots\pi_{i_m}\text{ has relative order }\tau\}. \] For integers $r,n\ge0$, put \[ \psi_{\tau,r}(n)=\#\{\pi\in S_n:\operatorname{occ}_{\tau}(\pi)=r\}, \qquad \operatorname{Av}_n(\tau)=\{\pi\in S_n:\operatorname{occ}_{\tau}(\pi)=0\}. \] Thus $\psi_{\tau,0}(n)=|\operatorname{Av}_n(\tau)|$. Two patterns are Wilf-equivalent when their avoidance counts agree for every $n$. This definition alone does not identify their counts with a positive prescribed number of occurrences. The supplied record does not explicitly specify the occurrence convention; the definitions above make the standard interpretation explicit.

The asymptotic question concerns two unspecified Wilf classes of patterns of length four. For a pattern $\tau$ representing either intended class and each fixed integer $r\ge0$, the supplied conjectural formula is \[ \psi_{\tau,r}(n)\sim C_{\tau,r}\,n^r\psi_{\tau,0}(n) \qquad(n\to\infty). \] Here the usual positive-constant interpretation would require a constant $C_{\tau,r}>0$, independent of $n$, such that \[ \lim_{n\to\infty}\frac{\psi_{\tau,r}(n)}{n^r\psi_{\tau,0}(n)}=C_{\tau,r}. \] The source writes $C_r$; the additional pattern subscript records dependence that cannot be excluded from the supplied evidence. The two classes, their representatives, the intended range of $r$, and any invariance of positive-occurrence counts within a class are not supplied. Consequently, this is a conditional reconstruction of the intended question, not a conjecture asserted here for every pattern of length four.

For the moment-sequence component, define the formal series \[ F_{\tau,r}(x)=\sum_{n\ge0}\psi_{\tau,r}(n)x^n, \qquad g_{\tau,n}(q)=\sum_{k=0}^{\binom{n}{m}}q^k\psi_{\tau,k}(n), \qquad G_{\tau}(x;q)=\sum_{n\ge0}g_{\tau,n}(q)x^n. \] Here $\binom{n}{m}=0$ for $n<m$. The expression for $g_{\tau,n}(q)$ is the natural reconstruction of the damaged displayed sum in the record; the variable $x$ has been introduced to distinguish the ordinary generating-function variable from the occurrence weight $q$.

A real sequence $(a_n)_{n\ge0}$ is a Stieltjes moment sequence if there is a nonnegative Borel measure $\mu$ on $[0,\infty)$, with every moment finite, such that \[ a_n=\int_{[0,\infty)}t^n\,\mathrm{d}\mu(t)\qquad(n\ge0). \] The supplied material raises moment-sequence questions about the fixed-occurrence coefficient sequences and about the weighted sequence $(g_{\tau,n}(q))_{n\ge0}$. However, the excerpt ends immediately after introducing the weighted coefficients: it supplies neither a complete conjecture about them nor a domain for $q$. No definite universally quantified moment-sequence assertion can therefore be recovered.

The excerpt attributes to Bostan, Elvey Price, Guttmann and Maillard a Stieltjes-moment result for the coefficients of all pattern-avoiding-permutation generating functions described there as known. It also reports increasing Hankel determinants for $\operatorname{Av}(1324)$ as the determinant size increases, motivating a conjecture that its counting sequence is a Stieltjes moment sequence. The determinant indexing is unspecified. For positive fixed occurrence counts, it reports failure of the Stieltjes moment property for every pattern and every $r>0$ investigated; this is a qualification about the investigated cases, not a universal theorem. The weighted construction is attributed to Blitvic and Steingrímsson.

Short English statement

For each of two unidentified length-four pattern classes, the proposed asymptotic says that prescribing a fixed number $r$ of occurrences multiplies the avoidance count by approximately a constant times $n^r$. The record also discusses Stieltjes moment properties of fixed-occurrence counts and occurrence-weighted counts. Missing class identities and an incomplete moment-sequence proposal prevent a fully specified statement. The supplied open-status assessment is dated curation and requires independent review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005320$, title: Wilf Class Asymptotics with Fixed Pattern Occurrences; exact source problem code: catalogue entry. The registry labels the problem open and the statement corrected_verified; its verification note describes repairs to delimiters, notation, grammar and typography, rather than independent mathematical verification.
• Supplied report citation: Enumerative Combinatorics (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/57, pp. 3215--3305. The registry gives proposed year 2022 and no named proposer. Report link, also supplied as a literature source labelled "original OWR report": https://doi.org/10.4171/owr/2022/57.
• Supplied literature source labelled "Conway--Guttmann primary article": https://www.combinatorics.org/ojs/index.php/eljc/article/download/v32i1p3/pdf. The supplied assessment describes a 2025 study in which the asymptotic for both intended length-four Wilf classes is explicitly conjectural. No article title or fuller bibliographic details are supplied.
• Literature check date: 2026-08-22. The supplied assessment states that the asymptotic remains open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The excerpt attributes the avoidance moment-sequence discussion to Bostan, Elvey Price, Guttmann and Maillard, and the weighted construction to Blitvic and Steingrímsson. Separate authoritative citations for these attributed contributions are not supplied.