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

Joseph Myers jsm at polyomino.org.uk
Sun Jul 12 01:42:03 CEST 2020

On Sat, 11 Jul 2020, Richard J. Mathar wrote:

> 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?

If you draw the short diagonals of each tile in the rhombille tiling, you 
get the edges of the regular hexagonal tiling (which are the same as edges 
in the triangular lattice with angles constrained to 120 degrees), and two 
edges are adjacent if and only if the corresponding rhombi were adjacent.

Joseph S. Myers
jsm at polyomino.org.uk

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