Applying the Pólya-Burnside Enumeration Theorem

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

The Pólya–Burnside enumeration theorem is an extension of the Pólya–Burnside lemma, Burnside's lemma, the Cauchy–Frobenius lemma, or the orbit‐counting theorem.


Given a finite group acting on a set of elements, the Pólya–Burnside enumeration theorem counts the number of elements of a given type as a function of their order.

In this Demonstration, a set of binary strings of a given length is acted upon by the group . The first component acts by word-reversing, while the second acts by bit‐wise negation. Rewriting rules and corresponding orbits are explicitly worked out for these reflections.

The number of orbits is for even and for odd .


Contributed by: Hector Zenil and Oleksandr Pavlyk (March 2011)
Open content licensed under CC BY-NC-SA



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.