How do we count set partitions? This video explains set partitions and the combinatorics behind them. You'll learn how to count the number of ways to partition a set of n elements according to a specific integer partition.
This video is also a continuation of my previous video, Faa di Bruno's formula. For useful background information, I suggest you check out my previous video: https://youtu.be/x6HFgHLeL3c
Partitions of sets are a concept in discrete mathematics and relevant to many counting problems. It surprisingly appears in many areas of math, even continuous math, such as in repeated differentiation. A partition of a set of n elements into k nonempty subsets, means to group the n items of a set into some number, k groups such that no group is empty. For example, if we have 2 items, we can put them together in one group, or each item in a separate group of its own.
Here are some links for more information:
https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula
https://en.wikipedia.org/wiki/Bell_polynomials
https://en.wikipedia.org/wiki/Partition_of_a_set
Recommended Books:
******************************** Hypergraph Theory ********************************
"Hypergraph Theory: An Introduction": https://amzn.to/48WKqfy
******************************** Graph Theory ********************************
"Introduction to Graph Theory (Trudeau)": https://amzn.to/48ZWhtj
"Graph Theory (Diestel)": https://amzn.to/4aYCSdW
******************************** Misc. Undergraduate Mathematics ********************************
Discrete Mathematics with Applications (Epp): https://amzn.to/4aWC1dM
A Book of Abstract Algebra (Pinter): https://amzn.to/3S2QmfV
Language, Proof and Logic: https://amzn.to/47EIZkE
Linear Algebra and Its Applications: https://amzn.to/48QsoMt
All the Math You Missed: https://amzn.to/3u5dORP
These are my Amazon Affiliate links. As an Amazon Associate I may earn commissions for purchases made through the links above.
00:00 Stirling Number Review
03:00 Integer Partitions of Sets
04:30 Formula
05:27 Example 1
05:54 Example 2
07:00 Example 3
08:50 Formula Derivation
09:40 Intuition
#combinatorics