The Pólya enumeration theorem is a very general theorem that allows the number of discrete combinatorial objects of a given type to be enumerated (counted) as a
function of their "order." The most common application is in the counting
of the number of simple graphs of 
nodes, tournaments on 
nodes, trees and rooted trees
with 
branches, groups of order 
, etc. The theorem is an extension of the Cauchy-Frobenius
lemma.
Pólya Enumeration Theorem
See also
Cauchy-Frobenius Lemma, Cycle Index, Group, Polyhedron Coloring, Rooted Tree, Simple Graph, Tournament, TreeExplore with Wolfram|Alpha
References
Harary, F. "The Number of Linear, Directed, Rooted, and Connected Graphs." Trans. Amer. Math. Soc. 78, 445-463, 1955.Harary, F. "Pólya's Enumeration Theorem." Graph Theory. Reading, MA: Addison-Wesley, pp. 180-184, 1994.Harary, F. and Palmer, E. M. "Pólya's Theorem." Ch. 2 in Graphical Enumeration. New York: Academic Press, pp. 33-50, 1973.Pólya, G. "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen." Acta Math. 68, 145-254, 1937.Roberts, F. S. Applied Combinatorics. Englewood Cliffs, NJ: Prentice-Hall, 1984.Skiena, S. "Polya's Theory of Counting." §1.2.6 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 25-26, 1990.Tucker, A. Applied Combinatorics, 3rd ed. New York: Wiley, 1995.Referenced on Wolfram|Alpha
Pólya Enumeration TheoremCite this as:
Weisstein, Eric W. "Pólya Enumeration Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolyaEnumerationTheorem.html