I’ve uploaded a new version of my Math.Combinatorics.Multiset library (see the previous announcement here). I’ve added a few more fairly simple algorithms (splitting a multiset into two pieces in all possible ways; finding all size-k submultisets of a multiset), and one which is decidedly NOT simple. I wanted to generate all distinct *cyclic arrangements* of the elements from a multiset — in other words, sequences which are distinct up to cyclic rotations. For example, the multiset has six distinct cyclic arrangements:

How to generate all such cyclic arrangements (also called “necklaces”), given a multiset? I thought about it for a whole day and decided that it was rather difficult! (If you don’t believe me, try coding it yourself.) A bit of searching confirmed my suspicion but turned up a nice 2003 paper by Joe Sawada with a fast algorithm (which I still don’t quite understand — it depends on some non-trivial properties of necklaces!). So, I implemented it, as `Math.Combinatorics.Multiset.cycles`

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I’ve spent some time trying to understand the efficient algorithms to generate necklaces, although limited to sets, not multisets. I am very interested in hearing any explanations of the algorithm that you come up with. :)

There is some interesting stuff in the paper you’ve shared. is it possible to see your implementation?

The package containing the code is freely available here: http://hackage.haskell.org/package/multiset%2Dcomb In particular you can see the implementation of Sawada’s algorithm here: http://hackage.haskell.org/packages/archive/multiset-comb/0.2.2/doc/html/src/Math-Combinatorics-Multiset.html#cycles

Thanks!!