pith. sign in

arxiv: 2412.03525 · v3 · submitted 2024-12-04 · 🧮 math.CO

Pin classes II: Small pin classes

Robert Brignall , Ben Jarvis This is my paper

Pith reviewed 2026-05-23 07:36 UTC · model grok-4.3

classification 🧮 math.CO
keywords pin classespin permutationsgrowth ratesphase transitionpermutation classesperiodic structureenumeration
0
0 comments X

The pith

Pin classes have a phase transition at growth rate μ≈3.28277, with only countably many below it and uncountably many at it.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies pin classes, formed as all finite subpermutations inside a single infinite pin permutation. It proves that these classes display a sharp change in cardinality at growth rate μ≈3.28277. Below μ only countably many distinct classes exist, while exactly at μ the number becomes uncountable. The proof proceeds by showing that every class with growth rate less than μ is generated by a pin permutation whose pattern repeats periodically. This periodic restriction makes the growth rates below μ fully classifiable.

Core claim

Pin classes are permutation classes consisting of all finite subpermutations contained in an infinite pin permutation. There is a phase transition at μ≈3.28277: uncountably many different pin classes have growth rate exactly μ, yet only countably many have smaller growth rates. All pin classes with growth rate less than μ are essentially defined by pin permutations that possess a periodic structure, which yields a classification of the set of growth rates of pin classes up to μ.

What carries the argument

The threshold μ≈3.28277 together with the periodic structure of the pin permutations that generate all classes below it.

If this is right

  • The growth rates attained by pin classes below μ form a countable, explicitly describable set.
  • Every pin class below μ arises from a pin permutation whose entries repeat in a periodic pattern.
  • Exactly at μ the collection of distinct pin classes becomes uncountable.
  • The periodic structure supplies a concrete method for enumerating and distinguishing all pin classes whose growth rate is smaller than μ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same periodic-structure technique might classify growth rates in other families of permutation classes built from infinite objects.
  • The countable list of rates below μ could be turned into an algorithm that decides, for a given pin permutation, whether its class lies below the threshold.
  • The jump to uncountably many classes at μ may mark the point where well-quasi-ordering properties of pin permutations cease to hold in a uniform way.

Load-bearing premise

All pin classes with growth rate less than μ are essentially defined by pin permutations that possess a periodic structure.

What would settle it

An explicit pin class whose growth rate is strictly less than μ but whose generating infinite pin permutation has no periodic structure.

Figures

Figures reproduced from arXiv: 2412.03525 by Ben Jarvis, Robert Brignall.

Figure 1
Figure 1. Figure 1: A pin permutation whose basic encoding is the word 1lurdrdluldrurd, and whose memory encoding is ur luulrudrrddr ldul ludlrdurrudr . The memory encoding is more amenable than the basic encoding to handling subpermutations of pin permutations. For example, consider the permutation π = 4731526, with basic encoding e(π) = 1urulur and memory encoding m(π) = ruurruur luul ru. The permutation 25314 can be obtain… view at source ↗
Figure 2
Figure 2. Figure 2: The start of the infinite permutation corresponding to the word w1,2. We begin with the cases in which a pin sequence visits both quadrants 1 and 2 infinitely often, and infinitely many of those visits in each quadrant contain at least three points. Proposition 4.7. Let w be a pin sequence which visits only quadrants 1 and 2, and in which both rur and lul are recurrent factors. Then gr(Cw) ě ν2,2 « 3.39752… view at source ↗
read the original abstract

Pin permutations play an important role in the structural study of permutation classes, most notably in relation to simple permutations and well-quasi-ordering, and in enumerative consequences arising from these. In this paper, we continue our study of pin classes, which are permutation classes that comprise all the finite subpermutations contained in an infinite pin permutation. We show that there is a phase transition at $\mu\approx 3.28277$: there are uncountably many different pin classes whose growth rate is equal to $\mu$, yet only countably many below $\mu$. Furthermore, by showing that all pin classes with growth rate less than $\mu$ are essentially defined by pin permutations that possess a periodic structure, we classify the set of growth rates of pin classes up to $\mu$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript continues the study of pin classes (permutation classes consisting of all finite subpermutations contained in an infinite pin permutation). It establishes a phase transition at growth rate μ ≈ 3.28277: there are only countably many pin classes with growth rate strictly less than μ, while uncountably many attain exactly μ. The classification of growth rates below μ is obtained by proving that every such pin class arises from a pin permutation with periodic structure, via exhaustive case analysis on pin sequences and their substitution decompositions.

Significance. If the central claims hold, the result supplies a sharp cardinality transition in the set of attainable growth rates for pin classes together with an explicit classification below the threshold. The exhaustive case analysis establishing the periodicity criterion constitutes a concrete combinatorial contribution to the structural and enumerative theory of permutation classes.

minor comments (1)
  1. The numerical approximation μ ≈ 3.28277 is stated without an accompanying exact algebraic expression or reference to its derivation; if such an expression appears later in the text, it should be cross-referenced already in the abstract and introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of the phase transition at μ and the classification of growth rates below the threshold, and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via case analysis

full rationale

The central result (phase transition at μ with countability below and uncountability at μ) rests on an exhaustive case analysis of pin sequences and substitution decompositions proving that growth rate < μ implies periodic pin permutation structure. This is established directly in the manuscript rather than by reduction to fitted parameters, self-citations, or prior ansatzes. No load-bearing step equates a claimed prediction to its own inputs by construction. Self-citation to Paper I is present but not required for the key periodicity criterion or growth-rate classification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard combinatorial definitions of permutation classes, growth rates, and pin permutations; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Growth rate of a permutation class is defined as the limit superior of the nth root of the number of length-n members.
    Invoked to state the phase transition and the classification.
  • domain assumption Pin permutations are infinite sequences whose finite subpermutations generate a pin class.
    Central definition used throughout the abstract.

pith-pipeline@v0.9.0 · 5650 in / 1363 out tokens · 25198 ms · 2026-05-23T07:36:57.910249+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 2 internal anchors

  1. [1]

    On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern.Electron

    Richard Arratia. On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern.Electron. J. Combin., 6:Note 1, 4 pp., 1999.doi:10.37236/1477

    work page doi:10.37236/1477 1999

  2. [2]

    An algo- rithm for deciding the finiteness of the number of simple permutations in permutation classes.Adv

    Fr ´ed´erique Bassino, Mathilde Bouvel, Adeline Pierrot, and Dominique Rossin. An algo- rithm for deciding the finiteness of the number of simple permutations in permutation classes.Adv. in Appl. Math., 64:124–200, 2015.doi:10.1016/j.aam.2014.12.001

    work page doi:10.1016/j.aam.2014.12.001 2015

  3. [3]

    Enumeration of pin- permutations.Electron

    Fr ´ed´erique Bassino, Mathilde Bouvel, and Dominique Rossin. Enumeration of pin- permutations.Electron. J. Combin., 18(1):Paper 57, 39, 2011.doi:10.37236/544

    work page doi:10.37236/544 2011

  4. [4]

    Growth rates of geometric grid classes of permutations.Electron

    David Bevan. Growth rates of geometric grid classes of permutations.Electron. J. Combin., 21(4): Paper 51, 17pp, 2014.doi:10.37236/4834

    work page doi:10.37236/4834 2014

  5. [6]

    URL:http://arxiv.org/abs/1506.06673

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [7]

    Intervals of permutation class growth rates.Combinatorica, 38(2):279–303, Apr 2018.doi:10.1007/s00493-016-3349-2

    David Bevan. Intervals of permutation class growth rates.Combinatorica, 38(2):279–303, Apr 2018.doi:10.1007/s00493-016-3349-2

    work page doi:10.1007/s00493-016-3349-2 2018

  7. [8]

    Uncountably many enumerations of well-quasi-ordered permutation classes

    R. Brignall and V . Vatter. Uncountably many enumerations of well-quasi-ordered permu- tation classes.Combin. Theory,accepted. URL:https://arxiv.org/abs/2211.12397

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [9]

    Decomposing simple permuta- tions, with enumerative consequences.Combinatorica, 28:385–400, 2008.doi:10.1007/ s00493-008-2314-0

    Robert Brignall, Sophie Huczynska, and Vincent Vatter. Decomposing simple permuta- tions, with enumerative consequences.Combinatorica, 28:385–400, 2008.doi:10.1007/ s00493-008-2314-0

    work page 2008

  9. [10]

    Simple permutations: decidability and unavoidable substructures.Theoret

    Robert Brignall, Nik Ru ˇskuc, and Vincent Vatter. Simple permutations: decidability and unavoidable substructures.Theoret. Comput. Sci., 391(1–2):150–163, 2008.doi:10.1016/ j.tcs.2007.10.037

    work page 2008

  10. [11]

    Cartier and D

    P . Cartier and D. Foata.Probl` emes combinatoires de commutation et r´ earrangements. Lec- ture Notes in Mathematics, No. 85. Springer-Verlag, Berlin, 1969.doi:10.1007/ BFb0079468

    work page 1969

  11. [12]

    Factor complexity

    Julien Cassaigne and Franc ¸ois Nicolas. Factor complexity. InCombinatorics, automata and number theory, volume 135 ofEncyclopedia Math. Appl., pages 163–247. Cambridge Univ. Press, Cambridge, 2010.doi:10.1017/CBO9780511777653.005. 24

    work page doi:10.1017/cbo9780511777653.005 2010

  12. [13]

    Unavoidable induced subgraphs in large graphs with no homogeneous sets.J

    Maria Chudnovsky, Ringi Kim, Sang-il Oum, and Paul Seymour. Unavoidable induced subgraphs in large graphs with no homogeneous sets.J. Combin. Theory Ser. B, 118:1–12, 2016.doi:10.1016/j.jctb.2016.01.008

    work page doi:10.1016/j.jctb.2016.01.008 2016

  13. [14]

    TheX-class and almost-increasing permutations.Ann

    Sergi Elizalde. TheX-class and almost-increasing permutations.Ann. Comb., 15:51–68, 2011.doi:10.1007/s00026-011-0082-9

    work page doi:10.1007/s00026-011-0082-9 2011

  14. [15]

    Cambridge University Press, Cambridge, 2009.doi:10.1017/CBO9780511801655

    Philippe Flajolet and Robert Sedgewick.Analytic combinatorics. Cambridge University Press, Cambridge, 2009.doi:10.1017/CBO9780511801655

    work page doi:10.1017/cbo9780511801655 2009

  15. [16]

    Pytheas Fogg.Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics

    N. Pytheas Fogg.Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.doi:10.1007/b13861

    work page doi:10.1007/b13861 2002

  16. [17]

    Grid classes and the Fibonacci dichotomy for re- stricted permutations.Electron

    Sophie Huczynska and Vincent Vatter. Grid classes and the Fibonacci dichotomy for re- stricted permutations.Electron. J. Combin., 13: Paper No. 54, 14pp, 2006.doi:10.37236/ 1080

    work page 2006

  17. [18]

    B. Jarvis. Pin classes I: Growth rates and bounds. Submitted

    work page

  18. [19]

    On growth rates of closed permutation classes.Electron

    Tom ´aˇs Kaiser and Martin Klazar. On growth rates of closed permutation classes.Electron. J. Combin., 9(2): Paper No. 10, 20pp, 2003.doi:10.37236/1682

    work page doi:10.37236/1682 2003

  19. [20]

    Marston Morse and Gustav A. Hedlund. Symbolic Dynamics.Amer. J. Math., 60(4):815– 866, 1938.doi:10.2307/2371264

    work page doi:10.2307/2371264 1938

  20. [21]

    Growth rates of permutation classes: categorization up to the uncountability threshold.Israel J

    Jay Pantone and Vincent Vatter. Growth rates of permutation classes: categorization up to the uncountability threshold.Israel J. Math., 236(1):1–43, 2020.doi:10.1007/ s11856-020-1964-5

    work page 2020

  21. [22]

    Permutation classes of every growth rate above 2.48188.Mathematika, 56(1):182–192, 2010.doi:10.1112/S0025579309000503

    Vincent Vatter. Permutation classes of every growth rate above 2.48188.Mathematika, 56(1):182–192, 2010.doi:10.1112/S0025579309000503

    work page doi:10.1112/s0025579309000503 2010

  22. [23]

    Small permutation classes.Proc

    Vincent Vatter. Small permutation classes.Proc. Lond. Math. Soc. (3), 103:879–921, 2011. doi:10.1112/plms/pdr017

    work page doi:10.1112/plms/pdr017 2011

  23. [24]

    Permutation classes

    Vincent Vatter. Permutation classes. InHandbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), pages 753–833. CRC Press, Boca Raton, FL, 2015.doi:10. 1201/b18255

    work page 2015

  24. [25]

    Growth rates of permutation classes: from countable to uncountable.Proc

    Vincent Vatter. Growth rates of permutation classes: from countable to uncountable.Proc. Lond. Math. Soc. (3), 119(4):960–997, 2019.doi:10.1112/plms.12250

    work page doi:10.1112/plms.12250 2019

  25. [26]

    PhD thesis, Univ

    Steve Waton.On Permutation Classes Defined by Token Passing Networks, Gridding Matrices and Pictures: Three Flavours of Involvement. PhD thesis, Univ. of St Andrews, 2007.doi: 10023/237. 25

    work page 2007