Star transposition Gray codes for multiset permutations

Given integers k ≥ 2 $k\ge 2$ and a 1 , … ${a}_{1},\ldots ,{a}_{k}\ge 1$ let ≔ ( ) ${\boldsymbol{a}}:= ({a}_{1},\ldots ,{a}_{k})$ n + ⋯ $n:= {a}_{1}+\cdots \,\,+{a}_{k}$ . An a-multiset permutation is string of length $n$ that contains exactly i ${a}_{i}$ symbols $i$ for each = $i=1,\ldots ,k$ In this work we consider the problem exhaustively generating all ${\boldsymbol{a}}$ -multiset permutations by star transpositions, is, in step, first entry transposed with any other distinct from one. This far-ranging generalization several known results. For example, it ${a}_{1}=\cdots \,={a}_{k}=1$ can be generated while combinations $k=2$ these operations if only they are balanced ${a}_{1}={a}_{2}$ ), positive case following middle levels theorem. To understand general, introduce parameter Δ − max { } ${\rm{\Delta }}({\boldsymbol{a}}):= n-2\max \{{a}_{1},\ldots ,{a}_{k}\}$ allows us to distinguish three different regimes problem. We show < 0 }}({\boldsymbol{a}})\lt 0$ then transposition Gray code does not exist. also construct such codes > }}({\boldsymbol{a}})\gt assuming exist }}({\boldsymbol{a}})=0$ present some partial Our proofs establish Hamilton-connectedness or Hamilton-laceability underlying flip graphs, answer cases recent conjecture Shen Williams. particular, prove graph Hamilton-laceable.