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bibkey: doughertyblissgalvanpolleyshuster2026enumerating authors: Robert Dougherty-Bliss, Alejandro B. Galvan, Michaela Polley, and David Shuster year: 2026 title: “Enumerating separable derangements” doi: null url: https://arxiv.org/abs/2608.27583v1 claim: “The paper gives an algorithm and asymptotic bounds for separable derangements, while Conjecture 15 leaves the limiting derangement proportion open.” strata_touched: [] license: citation-only triage: anchor

Enumerating separable derangements

Dougherty-Bliss, Galvan, Polley, and Shuster give a polynomial-time algorithm for the number b_n of separable derangements (Theorems 1 and 10). Theorems 12 and 13 give bounds for b_n / s_n and identify the exponential growth constant 3 + 2 * sqrt(2), where s_n counts separable permutations. These results partially answer Vatter’s Problem 4.4 and provide bounds relevant to Question 4.5.

The paper’s Conjecture 15 states that b_n / s_n tends to zero. Thus the limiting ratio remains a conjecture in this source, and numerical monotonicity or fitted asymptotics do not settle it. This note makes no claim about the separate largest-limit question following Vatter’s Question 4.3 and has no full formal D5 settlement attached to it. The source is released under CC BY 4.0; this repository note records citation and status only.

Equation (7) gives the actual separable-count asymptotic with exponential constant 3 + 2 * sqrt(2) and power n^(-3/2). This is a known input for fixed-shift count ratios, rather than a new settlement of Conjecture 15. Pinsky’s The Infinite Limit of Separable Permutations provides finite decomposition and scalar length limits. Transporting finite histories to an infinite law still requires the actual coupled transitions and absolute position/value labels.

The Brownian limit of separable permutations, Theorem 1.6, concerns normalized permuton convergence; an absolute fixed-point probability needs an additional argument. The recursive and Brownian separable permuton models are distinct, as shown by Féray and Rivera-López in their recursive separable permuton limit result, Proposition 1.3. A recursive sampler cannot replace the actual uniform class.

The official arXiv record remains v1. A bounded check of arXiv metadata and the related primary sources above identified no complete resolution of the full actual-class, all-length limit. Forward-citation coverage is incomplete; this is not a global originality certificate. The finite bridge in D5/S1/Words/Patterns/Separable/StableEndpointCodes.lean supplies bounded codes independent of the initial length, complete successful-history fibers, and exact fixed-family mass and absolute-coordinate cylinder sums under n > H*K + max(2*K, max(m,B)). Cap fibers retain their selected-leaf counts, including the threshold-satisfying four-of-twenty-two example. These finite statements do not settle the conjecture: actual count-ratio and sign-half interfaces, the infinite coupled law, truncation tails, occupation and hitting transfer remain needed.

Verified locator

  • arXiv: https://arxiv.org/abs/2608.27583v1
  • HTML: https://arxiv.org/html/2608.27583v1, Theorems 1, 10, 12, 13 and Conjecture 15
  • Submitted August 27, 2026 (version v1)

Actual occupation comparison

For positive length, let M_n(e) count actual sum-indecomposable avoiders with at least one point on displacement diagonal e, once per permutation. With b=sqrt(2)-1, rho=b^2 and a=1-b, the raw mass is C(e)=sum_(n>=1)rho^n M_n(e), while O(e)=(2/a)C(e) is its normalized version. The source diagonal counts are at-least-one events, rather than expected numbers of points.

OccupiedComparison.actual_occupation_subsolution_comparison proves summability of the actual shape weights, their total b, critical kernel mass 1/2, convergence of every occupied-count series, and the raw mass box 0<=C(e)<=a/2 without additional premises. The native Schroder recurrence, a uniform bound on nonnegative partial sums, and the Cauchy product supply the critical analytic facts; the singleton correction is retained.

The theorem also supplies the two actual finite occupied-count recurrences through the minimum-cut equivalence. The direct recurrence counts left-hit/right-no-hit and right-hit, and the skew recurrence adds disjoint factor events shifted by the opposite block lengths. These count each shape once, including all natural lengths and integer displacements. Splitting the actual hit fiber by the proper-cut predicate gives the full/indecomposable/ decomposable count partition. The native opposite-sign law identifies the decomposable hit counts with opposite-orientation indecomposable hit counts at lengths at least two. The actual singleton hit count is one on diagonal zero and zero elsewhere in either orientation.

Inversion preserves the avoidance pair and both proper-cut classes and negates every hit displacement, giving reflection of actual occupied counts. The absolutely convergent joint sums transport the finite recurrences with their actual shifts and intersection correction. Writing T for full occupied mass, P for direct-decomposable occupied mass and V(x)=x squared/(h+x), the direct recurrence gives T=2C-2V(C) and P=C-2V(C). Substitution in the skew recurrence and averaging its reflected version establishes the literal nonlinear occupation equation, including the forcing and singleton correction. Global comparison therefore has no actual fixed-point premise.

The required positive logarithmic subsolution is not constructed. The uniform all-integer lower bound registered in issue 15063 and the source Conjecture 15 remain unproved by this result.