Uploaded July 2021 | Updated September 2026, 3 weeks ago

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In this video, I use a group action, Burnside's lemma, Stirling numbers of the first kind, and the rising factorial to prove the number of ways to create an unordered string drawing from an alphabet with replacement.

00:00 Intro
00:35 Explaining the problem
02:44 Introducing permutations
04:40 Cycle notation
06:40 Group action
08:12 Orbits
09:30 Fixed points
10:20 Burnside's Lemma
11:37 Number of fixed points for a permutation
15:12 Stirling Numbers of the first kind
16:32 Stirling number identity
20:28 Examples of Stirling polynomials
23:50 Rising factorial identity proof
28:14 Final answer

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