## Abstract

A flip-swap language is a set S of binary strings of length n such that S∪{0^{n}} is closed under two operations (when applicable): (1) Flip the leftmost 1; and (2) Swap the leftmost 1 with the bit to its right. Flip-swap languages model many combinatorial objects including necklaces, Lyndon words, prefix normal words, left factors of k-ary Dyck words, lattice paths with at most k flaws, and feasible solutions to 0-1 knapsack problems. We prove that any flip-swap language forms a cyclic 2-Gray code when listed in binary reflected Gray code (BRGC) order. Furthermore, a generic successor rule computes the next string when provided with a membership tester. The rule generates each string in the aforementioned flip-swap languages in O(n)-amortized per string, except for prefix normal words of length n which require O(n^{1.864})-amortized per string. Our work generalizes results on necklaces and Lyndon words by Vajnovszki (2008) [35].

Original language | English |
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Pages (from-to) | 138-148 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 933 |

DOIs | |

Publication status | Published - 14 Oct 2022 |

## Keywords

- BRGC
- Binary reflected Gray code
- Dyck words
- Flip-swap language
- Gray codes
- Lyndon words
- Necklaces