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On the permutations generated by cyclic shift. (English) Zbl 1217.05013

Summary: The set of permutations generated by cyclic shift is studied using a number system coding for these permutations. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code describing structural and symmetry properties of the set of permutations ordered according to generation by cyclic shift. The code is associated with an Hamiltonian cycle in a regular weighted digraph. This Hamiltonian cycle is conjectured to be of minimal weight, leading to a combinatorial Gray code listing the set of permutations.

MSC:

05A05 Permutations, words, matrices
11A63 Radix representation; digital problems
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles

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