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Pin classes II: Small pin classes

Robert Brignall, Ben Jarvis

TL;DR

This work identifies a phase transition in the growth-rate spectrum of pin classes at $\mu \approx 3.28277$, proving that uncountably many pin classes achieve $\mathrm{gr}=\mu$ while only countably many do below. It further shows that pin classes with $\mathrm{gr}<\mu$ are governed by pin permutations with essentially periodic structure, yielding growth rates restricted to $\kappa$ or $\nu_{\ell}$, where $\nu_{\ell}$ are defined as the largest real zeros of specific polynomials. The authors develop a toolkit—memory encoding for pin words, box-sum analysis with trace monoids, and word-combinatorics (including Sturmian sequences)—to construct uncountably many $\mu$-growth pin classes and to classify those below $\mu$. These results illuminate the role of oscillations and periodicity in the enumerative behavior of pin classes and contribute a detailed map of growth-rate boundaries in the permutation-class landscape.

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 $μ\approx 3.28277$: there are uncountably many different pin classes whose growth rate is equal to $μ$, yet only countably many below $μ$. Furthermore, by showing that all pin classes with growth rate less than $μ$ are essentially defined by pin permutations that possess a periodic structure, we classify the set of growth rates of pin classes up to $μ$.

Pin classes II: Small pin classes

TL;DR

This work identifies a phase transition in the growth-rate spectrum of pin classes at , proving that uncountably many pin classes achieve while only countably many do below. It further shows that pin classes with are governed by pin permutations with essentially periodic structure, yielding growth rates restricted to or , where are defined as the largest real zeros of specific polynomials. The authors develop a toolkit—memory encoding for pin words, box-sum analysis with trace monoids, and word-combinatorics (including Sturmian sequences)—to construct uncountably many -growth pin classes and to classify those below . These results illuminate the role of oscillations and periodicity in the enumerative behavior of pin classes and contribute a detailed map of growth-rate boundaries in the permutation-class landscape.

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 : there are uncountably many different pin classes whose growth rate is equal to , yet only countably many below . Furthermore, by showing that all pin classes with growth rate less than are essentially defined by pin permutations that possess a periodic structure, we classify the set of growth rates of pin classes up to .

Paper Structure

This paper contains 15 sections, 32 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pin permutations
  4. Box sums
  5. Box closed classes and enumeration
  6. Combinatorics of Words
  7. Pin class structure
  8. Visits
  9. Visits to two quadrants and periodic words
  10. Pin sequences from binary words
  11. A spectrum of growth rates
  12. Non-periodic classes in two quadrants
  13. Classification of pin classes below mu
  14. Concluding remarks
  15. Acknowledgements

Key Result

Lemma 2.1

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^\sharp$ denote the set of gridded permutations with some fixed numbers $k$ of horizontal and $\ell$ of vertical lines. Then $\overline{\mathrm{gr}}(\mathcal{C}) = \overline{\mathrm{gr}}(\mathcal{C}^\sharp)$ and $\underline{\mathrm{gr}}(\

Figures (2)

  • Figure 1: A pin permutation whose basic encoding is the word $1\mathsf{l}\mathsf{u}\mathsf{r}\mathsf{d}\mathsf{r}\mathsf{d}\mathsf{l}\mathsf{u}\mathsf{l}\mathsf{d}\mathsf{r}\mathsf{u}\mathsf{r}\mathsf{d}$, and whose memory encoding is $\mathsf{u}_\mathsf{r}\mathsf{l}_\mathsf{u}\mathsf{u}_\mathsf{l}\mathsf{r}_\mathsf{u}\mathsf{d}_\mathsf{r}\mathsf{r}_\mathsf{d}\mathsf{d}_\mathsf{r}\mathsf{l}_\mathsf{d}\mathsf{u}_\mathsf{l}\mathsf{l}_\mathsf{u}\mathsf{d}_\mathsf{l}\mathsf{r}_\mathsf{d}\mathsf{u}_\mathsf{r}\mathsf{r}_\mathsf{u}\mathsf{d}_\mathsf{r}$.
  • Figure 2: The start of the infinite permutation corresponding to the word $w_{1,2}$.

Theorems & Definitions (50)

  • Lemma 2.1: C.f. Vatter vatter:small-permutati:
  • proof
  • Lemma 2.3: Cartier and Foata cartier:problemes-combi:; see also flajolet:analytic-combin:
  • Corollary 2.4
  • Theorem 2.5: Jarvis jarvis:pin-classes-i
  • Proposition 2.6: See Cassaigne and Nicolas cassaigne:factor-complexity:
  • Theorem 2.7: Morse and Hedlund morse:symbolic-dynami:
  • Lemma 2.8: Essentially due to Morse and Hedlund morse:symbolic-dynami:
  • proof
  • Theorem 2.9
  • ...and 40 more