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Pattern-avoiding shallow permutations

Kassie Archer, Aaron Geary, Robert P. Laudone

TL;DR

This work develops a systematic study of pattern-avoiding shallow permutations by exploiting Diaconis–Graham bounds, symmetry operations, and recursive constructions. It proves exact enumerations for all patterns in $\mathcal{S}_3$ (132,213,231,312,123,321), often via generating functions, and reveals rich structure under descents and fixed-symmetry classes (involution, centrosymmetric, persymmetric). A striking unifying theme is that both 132- and 321-avoiding shallow permutations share the Fibonacci enumeration $t_n=F_{2n-1}$, with detailed symmetric refinements and explicit closed forms for many refined counts. The results illuminate the interplay between shallowness, pattern avoidance, and permutation symmetries, and open avenues for mesh-pattern characterizations and broader combinatorial connections with knots and direct-sum inflations.

Abstract

Shallow permutations were defined in 1977 to be those that satisfy the lower bound of the Diaconis-Graham inequality. Recently, there has been renewed interest in these permutations. In particular, Berman and Tenner showed they satisfy certain pattern avoidance conditions in their cycle form and Woo showed they are exactly those whose cycle diagrams are unlinked. Shallow permutations that avoid 321 have appeared in many contexts; they are those permutations for which depth equals the reflection length, they have unimodal cycles, and they have been called Boolean permutations. Motivated by this interest in 321-avoiding shallow permutations, we investigate $σ$-avoiding shallow permutations for all $σ\in \mathcal{S}_3$. To do this, we develop more general structural results about shallow permutations, and apply them to enumerate shallow permutations avoiding any pattern of length 3.

Pattern-avoiding shallow permutations

TL;DR

This work develops a systematic study of pattern-avoiding shallow permutations by exploiting Diaconis–Graham bounds, symmetry operations, and recursive constructions. It proves exact enumerations for all patterns in (132,213,231,312,123,321), often via generating functions, and reveals rich structure under descents and fixed-symmetry classes (involution, centrosymmetric, persymmetric). A striking unifying theme is that both 132- and 321-avoiding shallow permutations share the Fibonacci enumeration , with detailed symmetric refinements and explicit closed forms for many refined counts. The results illuminate the interplay between shallowness, pattern avoidance, and permutation symmetries, and open avenues for mesh-pattern characterizations and broader combinatorial connections with knots and direct-sum inflations.

Abstract

Shallow permutations were defined in 1977 to be those that satisfy the lower bound of the Diaconis-Graham inequality. Recently, there has been renewed interest in these permutations. In particular, Berman and Tenner showed they satisfy certain pattern avoidance conditions in their cycle form and Woo showed they are exactly those whose cycle diagrams are unlinked. Shallow permutations that avoid 321 have appeared in many contexts; they are those permutations for which depth equals the reflection length, they have unimodal cycles, and they have been called Boolean permutations. Motivated by this interest in 321-avoiding shallow permutations, we investigate -avoiding shallow permutations for all . To do this, we develop more general structural results about shallow permutations, and apply them to enumerate shallow permutations avoiding any pattern of length 3.

Paper Structure

This paper contains 18 sections, 35 theorems, 81 equations, 1 table.

Table of Contents

  1. Introduction and background
  2. Structure of shallow permutations
  3. Shallow permutations that avoid 132 or 213
  4. Enumeration of 132-avoiding shallow permutations
  5. 132-avoiding shallow permutation by descent number
  6. 132-avoiding shallow permutations with symmetry
  7. Shallow permutations that avoid 231 or 312
  8. Enumeration of 231-avoiding shallow permutations
  9. 231-avoiding shallow permutations by descent number
  10. 231-avoiding shallow permutations with symmetry
  11. Shallow permutations that avoid 123
  12. Enumeration of 123-avoiding shallow permutations
  13. 123-avoiding shallow permutations with symmetry
  14. Shallow permutations that avoid 321
  15. Enumeration of 321-avoiding shallow permutations
  16. ...and 3 more sections

Key Result

Theorem 2.1

HM13 Suppose $\pi \in \mathcal{S}_n$ and for $n\geq 2$, define Then $\pi =1 \in \mathcal{S}_1$ is shallow, and when $n\geq 2,$

Theorems & Definitions (71)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 61 more