Star transposition Gray codes for multiset permutations
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Given integers k≥ 2 and a_1,…,a_k≥ 1, let a:=(a_1,…,a_k) and n:=a_1+⋯+a_k. An a-multiset permutation is a string of length n that contains exactly a_i symbols i for each i=1,…,k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, i.e., 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 (a_1=⋯=a_k=1) can be generated by star transpositions, while combinations (k=2) can be generated by these operations if and only if they are balanced (a_1=a_2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Δ(a):=n-2max{a_1,…,a_k} that allows us to distinguish three different regimes for this problem. We show that if Δ(a)<0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Δ(a)>0, assuming that they exist for the case Δ(a)=0. For the case Δ(a)=0 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.
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