Inside the Binary Reflected Gray Code: Flip-Swap Languages in 2-Gray Code Order

A flip-swap language is a set \(\mathbf{S}\) of binary strings length n such that \(\mathbf{S} \cup \{0^n\}\) closed under two operations (when applicable): (1) Flip the leftmost 1; and (2) Swap 1 with bit to its right. Flip-swap languages model many combinatorial objects including necklaces, Lyndon words, prefix normal left factors k-ary Dyck feasible solutions 0-1 knapsack problems. We prove any forms cyclic 2-Gray code when listed in reflected Gray (BRGC) order. Furthermore, generic successor rule computes next string provided membership tester. The generates each aforementioned O(n)-amortized per string, except for words which require \(O(n^{1.864})\)-amortized string. Our work generalizes results on necklaces by Vajnovski [Inf. Process. Lett. 106(3):96−99, 2008].