Enumeration of Chord Diagrams without Loops and Parallel Chords
  We enumerate chord diagrams without loops and without both loops and parallel chords. We show that the former ones describe Hamiltonian paths in $n$-dimensional octahedrons. The latter ones are also known as shapes. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations. 
Evgeniy Krasko and Alexander Omelchenko
math.CO