[seqfan] relation of rhombilles in A197459 and polyedges in A159867

[Richard J. Mathar]

Sequence A159867 counts (free) polyedges embedded
in the triangular lattice: connected sets of unit edges
that meet at angles of 60, 120 or 180 degrees. Rotation
or flips of an edge set do not generate a new edge set.

If we discard all of them which have (at least) one edge
of 60 or 180 degrees, only those remain that have
only angles of 120 degrees. These look like partially incomplete
benzene rings, and are illustrated in
https://www.mpia.de/~mathar/progs/A197459.pdf .
Counting these gives apparently A197459 (tested up to 1859 edge
sets). My question is basically: is there indeed a 1-to-1
bijection of these edge sets with the rhombille tilings
of that A197459 sequence, and how does that work?

Richard J. Mathar