Burnside Lemma Calculator for Symmetry Orbit Counting

Calculator Inputs

Scenario Name

Give this run a descriptive label.

Group Order

This must equal the number of symmetries entered.

Decimal Precision

Choose formatting for displayed results.

Symmetry Labels

One label per line. Leave blank for automatic labels.

Fixed-Point Counts

Use commas, spaces, or new lines between values.

Formula Used

Burnside’s Lemma counts distinct objects under a finite group action by averaging the number of configurations fixed by each group element.

Formula: Number of orbits = (1 / |G|) × Σ Fix(g)

Here, |G| is the group order, and Fix(g) is the number of configurations unchanged by symmetry g.

The calculator sums all fixed counts, divides by the group order, then reports the orbit total and descriptive summary statistics.

How to Use This Calculator

Enter a scenario name for your combinatorics problem.
Set the group order equal to the total number of symmetries.
List symmetry names line by line, or leave them blank.
Enter the fixed-point count for each symmetry.
Choose the decimal precision you want displayed.
Click Calculate Orbits to compute the Burnside average.
Review the result cards, contribution table, and Plotly chart.
Export the output as CSV or PDF for notes or teaching.

Example Data Table

Example: colorings of a square’s four corners with three colors under the dihedral group D₄.

Symmetry	Fixed Configurations	Reason
Identity	81	All 3⁴ colorings stay unchanged.
Rotate 90°	3	All corners must share one color.
Rotate 180°	9	Opposite corners must match in pairs.
Rotate 270°	3	Same condition as 90° rotation.
Reflect Vertical	27	Mirrored corners must match.
Reflect Horizontal	27	Mirrored corners must match.
Reflect Main Diagonal	27	Diagonal reflection creates pair constraints.
Reflect Other Diagonal	27	Diagonal reflection creates pair constraints.

Using the fixed counts above: (81 + 3 + 9 + 3 + 27 + 27 + 27 + 27) / 8 = 25.5. If instead you analyze vertex labeling assumptions with counts 81, 27, 9, 27, 27, 9, 27, 9, the orbit total becomes 216 / 8 = 27.

FAQs

1. What does Burnside’s Lemma calculate?

It counts distinct configurations after accounting for symmetry. Instead of listing every arrangement manually, it averages how many configurations remain fixed under each group action.

2. What is a fixed-point count?

A fixed-point count is the number of configurations left unchanged by one symmetry. Each rotation or reflection can preserve a different number of arrangements.

3. Why must the number of counts equal the group order?

Burnside’s Lemma averages over every group element. If one symmetry is missing, the average is incomplete and the orbit total becomes invalid.

4. Can I use decimals in the fixed counts?

You can enter decimals for exploratory work, but valid combinatorics problems usually produce whole-number fixed counts and an integer orbit result.

5. What if the orbit count is not an integer?

That usually signals incorrect fixed counts, a missing symmetry, or a group action modeled incorrectly. Review the assumptions and recompute each fixed set carefully.

6. Is this only for colorings?

No. It works for necklaces, tilings, graph labelings, pattern classes, board arrangements, and many other finite symmetry-counting problems.

7. What does the chart show?

The Plotly chart visualizes fixed configurations by symmetry. It helps you see which group elements contribute most to the Burnside average.

8. When should I use Pólya instead?

Use Pólya enumeration when you want a more general counting framework with cycle indices, multiple color weights, or symbolic generating functions.