Stirling Numbers
Count partitions and cycles.
Key idea
Count partitions and cycles.
In depth
Stirling numbers of the second kind S(n, k) count ways to partition n items into k non-empty unlabelled groups. They satisfy S(n, k) = kS(n − 1, k) + S(n − 1, k − 1).
Worked example
S(4, 2) = 7: ways to split {1, 2, 3, 4} into two non-empty groups.
Real-world uses
- Counting ways to split a group into teams and analysing algorithms that group items
Common mistake
Confusing with permutations, where order within groups would matter.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Stirling Numbers?
Count partitions and cycles. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Stirling Numbers used in real life?
Real-world examples: Counting ways to split a group into teams and analysing algorithms that group items.
What should I learn before Stirling Numbers?
Make sure you are comfortable with Catalan Numbers, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Stirling Numbers?
Confusing with permutations, where order within groups would matter.
How can I practise Stirling Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Stirling Numbers?
Move on to Partitions of Integers next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.