cp's OEIS Frontend

An n-isohedral coloring has n transitivity classes (or "orbits") of faces with respect to the color-preserving symmetry group of the coloring.

A n-isohedral "full coloring" requires that each face class uses a different color, so is therefore an n-colored, n-isohedral coloring. (The term "full coloring" appears to possibly be a new concept, not previously explored or documented in the theory of colorings.)

The full colorings (n colors) are arguably the most fundamental set of n-isohedral colorings -- even more so than the tilings (1 color) -- because all n-isohedral "sub-colorings" (1 to n-1 colors) can be derived from the n-isohedral full colorings.

In "Tilings and Patterns" by Branko Grünbaum and G.C. Shephard, 1986, pp. 102-107, the authors enumerate (as 32) the uniform edge-to-edge colorings of regular polygons (where uniform means that there is a single vertex transitivity type). Despite stating the requirement that "tiles in different transitivity classes... have different colors", they still include six colorings (three with triangles, and three with squares) in which two separate classes (within the coloring) use a single color. It is reasonable to assume they may have been referring to transitivity classes in the original uncolored tiling, not the coloring itself. However, this seems an arguably less consistent (and useful) treatment than enumerating only "full colorings", in which we refer to the face classes relative to the color-preserving symmetry group of the coloring itself.

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.