Star transposition Gray codes for multiset permutations

Abstract

AbstractGiven integers and , let and . An a‐multiset permutation is a string of length that contains exactly symbols for each . In this work we consider the problem of exhaustively generating all ‐multiset permutations by star transpositions, that is, in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far‐ranging generalization of several known results. For example, it is known that permutations () can be generated by star transpositions, while combinations () can be generated by these operations if and only if they are balanced (), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter that allows us to distinguish three different regimes for this problem. We show that if , then a star transposition Gray code for a‐multiset permutations does not exist. We also construct such Gray codes for the case , assuming that they exist for the case . For the case we present some partial positive results. Our proofs establish Hamilton‐connectedness or Hamilton‐laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton‐laceable.