A two-sided n -omino is a 1-by-n board of n squares with...

A two-sided $n$ -omino is a 1-by-n board of $n$ squares with each square $(2 n$ in in all because of the two sides) colored with one of $p$ given colors (squares on opposite sides may be colored differently). How many nonequivalent two-sided $n$ -ominoes are there?

Key Concepts

Combinatorial Enumeration of Colorings

This concept involves counting the total number of ways to assign colors to components of a structure. Here, each face (or square) of the board can be independently colored from among p colors, and so the basic count of colorings comes from the product rule of combinatorics where each independent choice multiplies. This same idea is applied broadly, whether counting colorings, labelings, or arrangements where multiple parts of an object are colored or otherwise assigned values.

Equivalence Under Symmetry

Often in combinatorial problems objects that appear different in a naive count are considered equivalent if they can be transformed into one another by a symmetry operation. In this context, two configurations are deemed equivalent if a reorientation (such as a reflection or rotation) can take one configuration to the other. Understanding and classifying these equivalence classes is essential for counting distinct or nonequivalent structures.

Group Actions and Burnside's Lemma

Group actions describe how a set of symmetries (such as reflections or rotations) can act on the collection of objects, partitioning them into orbits of equivalent configurations. Burnside's Lemma is a tool from group theory which simplifies counting the number of nonequivalent objects by averaging the number of configurations fixed by each element of the symmetry group. This approach is widely used to solve enumeration problems in combinatorics where symmetry plays an essential role.

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