Legendre, Stéphane; Paclet, Philippe On the permutations generated by cyclic shift. (English) Zbl 1217.05013 J. Integer Seq. 14, No. 3, Article 11.3.2, 14 p. (2011). 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 Keywords:permutations; cyclic shift; number system; palindrome; combinatorial gray code Software:OEIS PDFBibTeX XMLCite \textit{S. Legendre} and \textit{P. Paclet}, J. Integer Seq. 14, No. 3, Article 11.3.2, 14 p. (2011; Zbl 1217.05013) Full Text: arXiv EuDML EMIS Online Encyclopedia of Integer Sequences: The ruler function: exponent of the highest power of 2 dividing 2n. Equivalently, the 2-adic valuation of 2n. a(n) = Sum_{k=1..n} k!. a(n) = n! - 1. Ruler function associated with the set of permutations generated by cyclic shift, array read by rows. Ruler function associated with the set of permutations generated by cyclic shift in cyclic order, array read by rows.