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Generating Signed Permutations by Twisting Two-Sided Ribbons

Yuan, Qiu, Aaron Williams

TL;DR

The paper addresses the problem of efficiently generating all signed permutations of $[n]$, which number $2^n \cdot n!$, by introducing twisted plain changes—a twist-based generalization of plain changes implemented with two-sided ribbons. The authors develop a greedy construction that prioritizes $2$-twists on the largest symbol before $1$-twists, yielding a twist Gray code for $S_n^{\pm}$ and a loopless, mixed-radix Gray code implementation based on signed ruler sequences. Key contributions include the definition and analysis of twisted plain changes, the 2-twisted permutohedron framework, a signed ruler-sequence formalism, and an explicit loopless algorithm with a Python appendix. The work provides an efficient, memory-light method to enumerate signed permutations with potential impact in areas like genomics and network design, and it connects classical Gray codes to a rich signed-twist generalization through ruler-based encoding.

Abstract

We provide a simple and natural solution to the problem of generating all $2^n \cdot n!$ signed permutations of $[n] = \{1,2,\ldots,n\}$. Our solution provides a pleasing generalization of 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 swapping a pair of adjacent symbols, and the order is often visualized as a weaving pattern involving $n$ ropes. Here we model a signed permutation using $n$ ribbons with two distinct sides, and each successive configuration is created by twisting (i.e., swapping and turning over) two neighboring ribbons or a single ribbon. By greedily prioritizing $2$-twists of the largest symbol before $1$-twists of the largest symbol, we create a signed version of plain change's memorable zig-zag pattern. We provide a loopless algorithm (i.e., worst-case $\mathcal{O}(1)$-time per object) by extending the well-known mixed-radix Gray code algorithm.

Generating Signed Permutations by Twisting Two-Sided Ribbons

TL;DR

The paper addresses the problem of efficiently generating all signed permutations of , which number , by introducing twisted plain changes—a twist-based generalization of plain changes implemented with two-sided ribbons. The authors develop a greedy construction that prioritizes -twists on the largest symbol before -twists, yielding a twist Gray code for and a loopless, mixed-radix Gray code implementation based on signed ruler sequences. Key contributions include the definition and analysis of twisted plain changes, the 2-twisted permutohedron framework, a signed ruler-sequence formalism, and an explicit loopless algorithm with a Python appendix. The work provides an efficient, memory-light method to enumerate signed permutations with potential impact in areas like genomics and network design, and it connects classical Gray codes to a rich signed-twist generalization through ruler-based encoding.

Abstract

We provide a simple and natural solution to the problem of generating all signed permutations of . Our solution provides a pleasing generalization of the most famous ordering of permutations: plain changes (Steinhaus-Johnson-Trotter algorithm). In plain changes, the permutations of are ordered so that successive permutations differ by swapping a pair of adjacent symbols, and the order is often visualized as a weaving pattern involving ropes. Here we model a signed permutation using ribbons with two distinct sides, and each successive configuration is created by twisting (i.e., swapping and turning over) two neighboring ribbons or a single ribbon. By greedily prioritizing -twists of the largest symbol before -twists of the largest symbol, we create a signed version of plain change's memorable zig-zag pattern. We provide a loopless algorithm (i.e., worst-case -time per object) by extending the well-known mixed-radix Gray code algorithm.
Paper Structure (22 sections, 2 theorems, 5 equations, 7 figures, 1 table, 3 algorithms)

This paper contains 22 sections, 2 theorems, 5 equations, 7 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Generating Permutations
  3. Generating Signed Permutations
  4. Physical Model: Ribbons
  5. Outline
  6. Combinatorial Generation
  7. Gray Codes and Loopless Algorithms
  8. The Greedy Gray Code Algorithm
  9. Binary Reflected Gray Code
  10. Plain Changes
  11. Twisted Plain Changes
  12. $2$-Twisted Permutohedron
  13. Global Structure
  14. Ruler Sequences
  15. Unsigned Sequences
  16. ...and 7 more sections

Key Result

theorem thmcountertheorem

Algorithm alg:greedy creates a twist Gray code for signed permutations which starts at ${+}1 {+}2 {+}3 \cdots {+}n \in S_{n}^{\pm}$ and prioritizes $2$-twists of the largest value followed by $1$-twists of the largest value. The order is cyclic as the last signed permutation is ${-}1 {+}2 {+}3 \cdot

Figures (7)

  • Figure 1: Two-sided ribbons with distinct positive (i.e., glossy) and negative (i.e., matte) sides running in parallel. A $k$-twist reverses the order of $k$ neighboring ribbons and turns each of them over, as shown for (a) $k=1$ and (b) $k=2$.
  • Figure 2: Binary reflected Gray code using indistinct two-sided ribbons for $n=4$.
  • Figure 3: Plain changes $\mathsf{plain}(n)$ using distinct one-sided ribbons for $n=4$.
  • Figure 4: Twisted plain changes $\mathsf{twisted}(n)$ for $n=4$ up to its $25$th entry.
  • Figure 5: The permutohedron and the $2$-twisted permutohedron for $n=4$.
  • ...and 2 more figures

Theorems & Definitions (4)

  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem