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Generating Signed Permutations by Twisting Two-Sided Ribbons

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LATIN 2024: Theoretical Informatics (LATIN 2024)

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Abstract

We provide a simple approach to generating all \(2^n \cdot n!\) signed permutations of \([n] = \{1,2,\ldots ,n\}\). Our solution generalizes the most famous ordering of permutations: plain changes (Steinhaus-Johnson-Trotter algorithm). In plain changes, the n! permutations of [n] are ordered so that successive permutations differ by swap** a pair of adjacent symbols, and the order is often visualized as a weaving pattern on n ropes. Here we model a signed permutation as n ribbons with two distinct sides, and each successive configuration is created by twisting (i.e., swap** and turning over) two neighboring ribbons or a single ribbon. By greedily prioritizing 2-twists of large symbols then 1-twists of large symbols, we create a signed version of plain change’s memorable zig-zag pattern. We also provide a loopless implementation (i.e., worst-case \(\mathcal {O}(1)\)-time per object) by enhancing the well-known mixed-radix Gray code algorithm.

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Notes

  1. 1.

    Manufacturers refer to this type of ribbon as single face as only one side is polished.

  2. 2.

    If the permutation is stored in a BLL instead of an array, then loopless is possible [35].

  3. 3.

    The eponymous Gray code by Gray [10] also demonstrates Stigler’s law [32]: [9, 13].

  4. 4.

    Physically, a ribbon moves above or below its neighbor, but that is not relevant here.

  5. 5.

    \(\overleftarrow{s_1}\) and \(\overrightarrow{s_1}\) are omitted as they equal other swaps. In fact, the swaps are all jumps [12].

  6. 6.

    Unsigned ruler sequences are palindromes, so \(^R\)’s can be added to (8) to mirror (10).

  7. 7.

    Surprisingly, this sequence is not yet in the Online Encyclopedia of Integer Sequences, nor is the signed downstairs sequence \(\textsf{ruler}\pm (n{,}n{-}1{,}{.}{.}{.}{,}2) = \textsf{ruler}\pm (n{,}n{-}1{,}{.}{.}{.}{,}1)-1 = 1,2,-1,2,1,3,-1,-2,1,-2,-1,3,1,2,-1,2,1,3,-1,-2,1,-2,-1,4,\ldots \).

  8. 8.

    Negative indices give right-to-left access in Python. So the ruler entry -1 selects the last function fns[-1] = twist(p,q,n,1) (i.e., 2-twist n right). Notes: v=v is for binding; slice notation [-1::-1] reverses a list; indices are reversed from Sect. 4.

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Authors and Affiliations

  1. Williams College, Williamstown, MA, 01267, USA

    Yuan Qiu & Aaron Williams

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  1. Yuan Qiu
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Correspondence to Aaron Williams .

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Editors and Affiliations

  1. DIM-CMM, Universidad de Chile, Santiago, Chile

    José A. Soto

  2. Technical University of Munich, Munich, Germany

    Andreas Wiese

A Python Implementation

A Python Implementation

A loopless implementation of our signed plain change order \(\textsf{twisted}(n)\) in Python 3. Entries in the twisted ruler sequence \(\textsf{ruler}\pm (n,n{-}1,\ldots ,2,1,2,2,\ldots ,2)\) select the 2-twist or 1-twist (i.e., flip) to applyFootnote 8. Programs are available online [37].

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Qiu, Y., Williams, A. (2024). Generating Signed Permutations by Twisting Two-Sided Ribbons. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14578. Springer, Cham. https://doi.org/10.1007/978-3-031-55598-5_8

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  • DOI: https://doi.org/10.1007/978-3-031-55598-5_8

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