Stirling Numbers - The Magic Duo

Previous video for reference:    • The Dark Side of Pascal's Triangle #SoME4   What are the Stirling numbers and why should we care about them? This video covers both an algebraically motivated definition as well as a combinatorial definition. Then their recurrence relations are proved, which give a relatively fast computation method for calculating them. Despite clear parallels between Stirling numbers of the first kind and Stirling numbers of the second kind, their combinatorial classes seem very different. A brief overview of their combinatorial link is also discussed. Technical errors: 0:49 The falling and rising factorial are products, and should not have plus signs next to the ellipsis. 9:30 The equation should read x^3 - 2x^2 - x^2 + 2x Supplement material: Finite Calculus: A Tutorial for Solving Nasty Sums by David Gleich: https://www.cs.purdue.edu/homes/dglei... My Patreon:   / zhulimath   0:00 Introduction 0:41 Falling and rising factorials 1:59 Algebraic definition of the Stirling numbers 2:58 Introducing recurrences of the Stirling numbers 4:12 Combinatorial definition of Stirling numbers of the first kind 6:09 Proving definitions of Stirling numbers of the first kind align 9:55 Combinatorial definition of Stirling numbers of the second kind 10:38 Proving definitions of Stirling numbers of the second kind align 12:28 Stirling numbers of the first kind recurrence combinatorial proof 13:40 Stirling numbers of the second kind recurrence combinatorial proof 14:36 Stirling numbers are related, but what is the combinatorial link? 16:33 Conclusion 17:33 Outro Intro riff taken from: Nikolai Kapustin - 8 Concert Etudes, Op. 40: III. Toccatina Music Credit: Sayuri Hayashi Egnell / Lupi Anna Landstrom / Across the Lake Megan Wofford / Holding on to Hope William Benckert / Meant to Love Johannes Bornlof / Turtle Swim Gavin Luke / Forest Moon courtesy of www.epidemicsound.com