A268591 -id:A268591 - cp's OEIS frontend

A269630 Number of n-isohedral edge-to-edge colorings of regular polygons.

3, 49, 359, 2591, 15294, 115638

Offset: 1

Brian Galebach, Mar 01 2016

An n-isohedral coloring has n transitivity classes (or "orbits") of faces with respect to the color-preserving symmetry group of the coloring.
An n-isohedral coloring may use anywhere from 1 color (a tiling) to n colors (a "full coloring").
Two colorings are considered identical (and hence not counted twice) if one can be obtained from the other through some permutation or reassignment of colors.
In "Tilings and Patterns" by Branko Grünbaum and G. C. Shephard, 1986, pp. 102-107, the authors choose to require, in their enumeration of uniform colorings, that "tiles in different transitivity classes... have different colors." However, this is more restrictive than the most general definition of a coloring, or "colored tiling", given on p. 102, which states that "to each tile t of a given tiling T we assign one of a finite set of colors." Furthermore, some other studies of colorings actually require that some tiles in different classes share a single color (see the "Transitive perfect colorings" link below for an example). Hence, the enumerations in this sequence adhere solely to the most general coloring definition, with the only restriction being the prohibition of color permutations between colorings, as described in the preceding paragraph.

			The three 1-isohedral colorings are the regular tilings (triangles, squares, hexagons).
The 49 2-isohedral colorings comprise the 13 2-isohedral tilings given in D. Chavey, 1984, the corresponding "full coloring" version of each of those 13 tilings, where each uses two colors (one for each tile type); plus 7 2-isohedral colorings of triangles, 9 2-isohedral colorings of squares, and 7 2-isohedral colorings of hexagons.
The 359 3-isohedral colorings comprise the 29 3-isohedral tilings, 126 full colorings (which use three colors each), and 204 colorings that use two colors each. These 359 colorings are illustrated in the Facebook link given above.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns, 1986.

D. Chavey, Periodic Tilings and Tilings by Regular Polygons I, Thesis, 1984, pp. 165-172 gives the 13 2-isohedral edge-to-edge tilings of regular polygons. Each of these tilings corresponds to two 2-isohedral edge-to-edge colorings of regular polygons (the tiling itself, plus the analogous "full coloring").
Brian Galebach, n-Isohedral Edge-to-Edge Colorings of Regular Polygons, Facebook
Junmar Gentuya and René Felix, Transitive perfect colorings of the non-regular Archimedean tilings, arXiv:1507.05153 [math.GR], 2013, finds edge-to-edge colorings of regular polygons satisfying certain criteria. All of the colorings found in this paper require that tiles in multiple transitivity classes share colors.

The n-isohedral edge-to-edge colorings of regular polygons comprise:
The n-isohedral edge-to-edge tilings of regular polygons (A268184), which use the same color for all face classes (1 color);
The n-isohedral edge-to-edge "full colorings" of regular polygons (A268591), which use a different color for each face class (n colors); and
All n-isohedral edge-to-edge colorings of regular polygons using between 2 and n-1 colors (future sequence).

cp's OEIS Frontend

A269630 Number of n-isohedral edge-to-edge colorings of regular polygons.

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