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Efficient counting of permutation patterns via double posets

Joscha Diehl, Emanuele Verri

TL;DR

This work reframes permutation-pattern counting through strict double posets, showing that corner trees and permutations can be encoded as specific double-poset families. By establishing an ($ ext{Epi}$, $ ext{RegMono}$)-factorization in the category of double posets and a linear endomorphism translating double-poset occurrences to linear pattern counts, the authors generalize the Even-Zohar–Leng algorithm. They introduce the Tree$_{5/3}$ family, enabling $ ilde{ ext{O}}(n^{5/3})$ counting for twelve additional directions at level 5, and provide a practical counting procedure for corner-tree occurrences via vertex/edge recurrences and sum-tree data structures. The framework suggests a path to counting broader classes of permutations faster than naive approaches and links to recent pattern-tree generalizations. Open questions concern extending the speedup to all level-5 directions and deriving closed enumerative formulas for the poset families involved.

Abstract

Corner trees, introduced in "Even-Zohar and Leng, 2021, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms", allow for the efficient counting of certain permutation patterns. Here we identify corner trees as a subset of finite (strict) double posets, which we term twin-tree double posets. They are contained in both twin double posets and tree double posets, giving candidate sets for generalizations of corner tree countings. We provide the generalization of an algorithm proposed by Even-Zohar/Leng to a class of tree double posets, thereby enlarging the space of permutations that can be counted in O(n^{5/3}).

Efficient counting of permutation patterns via double posets

TL;DR

This work reframes permutation-pattern counting through strict double posets, showing that corner trees and permutations can be encoded as specific double-poset families. By establishing an (, )-factorization in the category of double posets and a linear endomorphism translating double-poset occurrences to linear pattern counts, the authors generalize the Even-Zohar–Leng algorithm. They introduce the Tree family, enabling counting for twelve additional directions at level 5, and provide a practical counting procedure for corner-tree occurrences via vertex/edge recurrences and sum-tree data structures. The framework suggests a path to counting broader classes of permutations faster than naive approaches and links to recent pattern-tree generalizations. Open questions concern extending the speedup to all level-5 directions and deriving closed enumerative formulas for the poset families involved.

Abstract

Corner trees, introduced in "Even-Zohar and Leng, 2021, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms", allow for the efficient counting of certain permutation patterns. Here we identify corner trees as a subset of finite (strict) double posets, which we term twin-tree double posets. They are contained in both twin double posets and tree double posets, giving candidate sets for generalizations of corner tree countings. We provide the generalization of an algorithm proposed by Even-Zohar/Leng to a class of tree double posets, thereby enlarging the space of permutations that can be counted in O(n^{5/3}).
Paper Structure (19 sections, 35 theorems, 107 equations, 43 figures, 2 algorithms)

This paper contains 19 sections, 35 theorems, 107 equations, 43 figures, 2 algorithms.

Table of Contents

  1. Corner trees: counting linear combinations of permutation patterns
  2. Corner trees and permutations as double posets
  3. Unrooting corner trees
  4. Double posets
  5. Counting permutations using double posets
  6. Switching between homomorphisms and embeddings
  7. Counting double poset morphisms when ${\mathtt{D}} = \Psi_{\mathcal{S}\leftarrow\mathfrak{S}}(\Pi)$
  8. Generalization of the $[\mathtt{3\,2\,1\,4}]$ algorithm
  9. Family of double posets for which the algorithm works: $\mathsf{Tree_{\,5/3}}$
  10. The algorithm for counting corner tree occurrences
  11. Algorithm to count occurrences of $\mathsf{Tree_{\,5/3}}$
  12. New directions at level 5
  13. Conclusion and outlook
  14. Open questions
  15. Supplementary material
  16. ...and 4 more sections

Key Result

Proposition 2.2

Define the map $\Psi_{\mathsf{SN} \leftarrow\mathsf{CT}}: \mathsf{CornerTrees} \to \mathsf{SNpoly}$ as follows. Given a corner tree $\mathsf{ct}=(\textup{V}(\mathsf{ct}),\textup{E}(\mathsf{ct}))$, construct $\Psi_{\mathsf{SN} \leftarrow\mathsf{CT}}(\mathsf{ct})$ by Then, the map $\Psi_{\mathsf{SN} \leftarrow\mathsf{CT}}$ is surjective, meaning that every SN polytree arises from some corner tree u

Figures (43)

  • Figure 1: Illustration of the conversions between SN polytrees and corner trees
  • Figure 2: Double posets represented as pairs of Hasse diagrams
  • Figure 3: Example of a twin tree double poset arising from an SN polytree
  • Figure 4: Pairs of Hasse diagrams
  • Figure 5: $\mathsf{West}$ and $\mathsf{South}$ posets
  • ...and 38 more figures

Theorems & Definitions (103)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6: Twin double poset
  • ...and 93 more