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Skipping Ropes: An Efficient Gray Code Algorithm for Generating Wiggly Permutations

Authors Vincent Pilaud , Aaron Williams

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • permutations
  • wiggly permutations
  • pattern avoidance
  • permutahedron
  • wigglyhedron
  • Hamilton path
  • flip graph
  • Gray code
  • combinatorial generation
  • generation algorithm

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Abstract

Wiggly permutations were introduced by Bapat and Pilaud (Wigglyhedron Mathematische Zeitschrift 2025). We positively answer one of their conjectures by finding a Hamilton path in the wiggly flip graph that is isomorphic to the wigglyhedron. Our path provides a Gray code in which successive wiggly permutations are obtained by a single jump or hop, meaning that one or two consecutive symbols move past some number of smaller symbols. The Gray code has a simple greedy description that produces a recursive zig-zag pattern reminiscent of plain changes for permutations. More broadly, our results extend Algorithm J and the series of papers on zig-zag languages initiated by Hartung, Hoang, Mütze and Williams (Combinatorial Generation via Permutation Languages SODA 2020). Finally, we use wiggly changes as the basis for an 𝒪(n)-time delay generation algorithm.

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Vincent Pilaud and Aaron Williams. Skipping Ropes: An Efficient Gray Code Algorithm for Generating Wiggly Permutations. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.46

Author Details

Vincent Pilaud
  • Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Spain
Aaron Williams
  • Department of Computer Science, Williams College, Williamstown, MA, USA

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