Disorders and Permutations

Authors Laurent Bulteau, Samuele Giraudo , Stéphane Vialette

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Laurent Bulteau
  • LIGM, Univ Gustave Eiffel, CNRS, F-77454 Marne-la-Vallée, France
Samuele Giraudo
  • LIGM, Univ Gustave Eiffel, CNRS, F-77454 Marne-la-Vallée, France
Stéphane Vialette
  • LIGM, Univ Gustave Eiffel, CNRS, F-77454 Marne-la-Vallée, France

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Laurent Bulteau, Samuele Giraudo, and Stéphane Vialette. Disorders and Permutations. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The additive x-disorder of a permutation is the sum of the absolute differences of all pairs of consecutive elements. We show that the additive x-disorder of a permutation of S(n), n ≥ 2, ranges from n-1 to ⌊n²/2⌋ - 1, and we give a complete characterization of permutations having extreme such values. Moreover, for any positive integers n and d such that n ≥ 2 and n-1 ≤ d ≤ ⌊n²/2⌋ - 1, we propose a linear-time algorithm to compute a permutation π ∈ S(n) with additive x-disorder d.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • Permutation
  • Algorithm


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