Diagonal Permuton Limits for Skew-Sum Avoidance

Definitions and mathematical statement

For an integer $m\geq 1$, write $[m]=\{1,\ldots,m\}$ and let $S_m$ be the set of permutations of $[m]$. Fix $\alpha\in S_m$. Its skew sum with the one-entry permutation is $\tau=\alpha\ominus 1\in S_{m+1}$, defined by \[ \tau(i)=\alpha(i)+1\quad(1\leq i\leq m),\qquad \tau(m+1)=1. \] A permutation $\pi\in S_n$ contains a pattern $\sigma\in S_k$ if there are indices $1\leq i_1<\cdots<i_k\leq n$ such that \[ \pi(i_a)<\pi(i_b)\quad\Longleftrightarrow\quad \sigma(a)<\sigma(b) \qquad(1\leq a,b\leq k). \] It avoids $\sigma$ if no such indices exist. Set \[ \operatorname{Av}_n(\sigma)=\{\pi\in S_n:\pi\text{ avoids }\sigma\}, \qquad \operatorname{Av}(\sigma)=\bigcup_{n\geq 1}\operatorname{Av}_n(\sigma). \] For $\tau=\alpha\ominus1$, these finite avoidance sets are nonempty: the increasing permutation avoids $\tau$.

A permuton is a Borel probability measure on $[0,1]^2$ whose two coordinate marginals are Lebesgue measure on $[0,1]$. The diagonal permuton $\mu_{\mathrm{diag}}$ is the distribution of $(U,U)$, where $U$ is uniform on $[0,1]$; equivalently, \[ \int f\,{\,\mathrm{d}}\mu_{\mathrm{diag}}=\int_0^1 f(t,t)\,{\,\mathrm{d}} t \quad\text{for every }f\in C([0,1]^2). \] For $\pi\in S_n$, define its empirical plot measure by \[ \nu_\pi=\frac1n\sum_{i=1}^n\delta_{(i/n,\pi(i)/n)}. \] Its finite marginals are discrete uniform measures; its weak limits as $n\to\infty$ have uniform marginals and hence are permutons.

Conjecture attributed in the supplied record to David Bevan. For every integer $m\geq1$ and every fixed $\alpha\in S_m$, let $\Pi_n$ be uniformly distributed on $\operatorname{Av}_n(\alpha\ominus1)$. Then \[ \nu_{\Pi_n}\xrightarrow[n\to\infty]{\mathrm{weakly\ in\ probability}}\mu_{\mathrm{diag}}. \] Explicitly, for every $f\in C([0,1]^2)$ and every $\varepsilon>0$, \[ \lim_{n\to\infty} \mathbb P\!\left( \left|\frac1n\sum_{i=1}^n f\!\left(\frac{i}{n},\frac{\Pi_n(i)}{n}\right) -\int_0^1 f(t,t)\,{\,\mathrm{d}} t\right|>\varepsilon \right)=0. \] The uniform sampling law and convergence in probability make explicit the interpretation suggested by the source's uniform-random rare-region discussion. The conjecture sentence itself does not explicitly specify either convention, so this interpretation requires human confirmation.

Supplied rare-region background and qualifications. For the relevant pattern $\tau=\alpha\ominus1$, write $\mathbb P_n^\tau$ for the uniform probability measure on $\operatorname{Av}_n(\tau)$. A point $(x,y)\in[0,1]^2$ is $\tau$-rare if, for every sequence $(i_n,j_n)\in[n]^2$ with \[ (i_n/n,j_n/n)\longrightarrow(x,y), \] there exists $r\in(0,1)$, possibly depending on that sequence, such that \[ \mathbb P_n^\tau\{\pi(i_n)=j_n\}=o(r^n) \qquad(n\to\infty). \] Let $R_\tau\subseteq[0,1]^2$ be the set of these points. The excerpt introduces this definition for a general pattern $\tau\in S_k$; its probability law presupposes that the relevant avoidance sets are nonempty. That condition holds for the family in the conjecture.

The excerpt attributes to Atapour and Madras, reference [1, Theorem 8.1], and to Madras and Yıldırım, reference [2, Theorem 1], the assertion that, whenever $\tau=\alpha\ominus1$ with $\alpha$ nonempty, $(1,0)$ has an open neighborhood contained in $R_\tau$. At this boundary point, the neighborhood is naturally read relative to $[0,1]^2$.

The excerpt also attributes to Madras and Yıldırım, reference [2, Theorem 17], a convex-hull assertion printed as \[ (x,y)\in[0,1]\setminus R_\tau \quad\Longrightarrow\quad \operatorname{conv}\{(x,y),(0,0),(0,1)\} \subseteq[0,1]\setminus R_\tau. \] This is retained only as a report of the supplied wording. Its ambient set is dimensionally inconsistent with the definition of $R_\tau$, and both the intended ambient square and the listed vertices require checking against the original theorem. No corrected version is assumed here, and this background assertion is not a hypothesis of the conjecture.

Short English statement

For each fixed nonempty permutation $\alpha$, choose a uniformly random permutation of size $n$ avoiding $\alpha\ominus1$. The conjecture says that its rescaled plot converges in probability to uniform mass on the increasing diagonal. This states the natural uniform-random interpretation of the supplied conjecture; its convergence convention and a malformed auxiliary rare-region theorem require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005770$, titled Diagonal Permuton Limits for Skew-Sum Avoidance; exact source problem code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2024.
• Mini-Workshop: Permutation Patterns (2024), Oberwolfach Reports, pp. 273--308, DOI 10.4171/owr/2024/6. Supplied source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2024/6.
• The supplied excerpt is headed Permuton Limits of Permutation Classes and Rare Regions, Justin M. Troyka, joint work with David Bevan. It refers to the target as Bevan's conjecture. The registry's separate proposed_by field is null, so that field supplies no additional proposer attribution.
• Supplied literature-source link labeled "rare-region paper": https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p13. The excerpt separately names Atapour and Madras as reference [1], citing Theorem 8.1, and Madras and Yıldırım as reference [2], citing Theorems 1 and 17. Full bibliographic entries and an explicit identification of which numbered reference the supplied link represents are absent from the record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the record "partially solved" and states that rare-region theorems give partial geometric constraints for patterns $\alpha\ominus1$, while Bevan's full diagonal-permuton conjecture remains open. This is supplied dated curation, not an independently verified literature assessment. The background expressly asks that time-sensitive status and cited claims be verified independently.
• The supplied registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly and extraction artifacts were removed. These are source-record assertions; this draft does not claim to have visited or independently verified either external source.