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Counting Permutation Patterns with Multidimensional Trees

Authors Gal Beniamini , Nir Lavee

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ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Pattern matching
Keywords
  • Pattern counting
  • patterns
  • permutations

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Abstract

We consider the well-studied pattern-counting problem: given a permutation π ∈ 𝕊_n and an integer k > 1, count the number of order-isomorphic occurrences of every pattern τ ∈ 𝕊_k in π.
Our first result is an 𝒪̃(n²)-time algorithm for k = 6 and k = 7. The proof relies heavily on a new family of graphs that we introduce, called pattern-trees. Every such tree corresponds to an integer linear combination of permutations in 𝕊_k, and is associated with linear extensions of partially ordered sets. We design an evaluation algorithm for these combinations, and apply it to a family of linearly-independent trees. For k = 8, we show a barrier: the subspace spanned by trees in the previous family has dimension exactly |𝕊₈| - 1, one less than required.
Our second result is an 𝒪̃(n^{7/4})-time algorithm for k = 5. This algorithm extends the framework of pattern-trees by speeding-up their evaluation in certain cases. A key component of the proof is the introduction of pair-rectangle-trees, a data structure for dominance counting.

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Gal Beniamini and Nir Lavee. Counting Permutation Patterns with Multidimensional Trees. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.24

Author Details

Gal Beniamini
  • The Hebrew University of Jerusalem, Israel
Nir Lavee
  • The Hebrew University of Jerusalem, Israel

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References

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