CO29 Stirling Numbers of First Kind vs Stirling Numbers of Second Kind. Getting one from the other

How are the Stirling Numbers of First Kind related to those of Second Kind? The answer is surprising, and the method of proof is delightful. We prove that the matrix of signed Stirling Numbers of the First Kind is the inverse of the matrix of Stirling Numbers of the Second Kind. The proof shows that the two matrices are the transposes of the two change of basis matrices for two specific bases for a vector space of polynomials. Subscribe ‪@Shahriari‬ for more in-depth math videos for undergraduates.
00:00 Introduction
00:29 Stirling Numbers of the first kind (CO27 What are the Stirling Numbers of...)
00:54 Stirling Numbers of the second kind (CO24 The Stirling Numbers of the Seco...)
01:07 Motivating Question
01:30 Lecture Outline
02:41 Stirling Numbers of the 1st kind: Definition, Notation, Recurrence Relation
04:27 Table of small Stirling Numbers of the 1st Kind
04:36 Stirling Numbers of the 2nd kind: Definition, Notation, Recurrence Relation
05:49 Table of small Stirling Numbers of the 2nd Kind
06:01 How are the Stirling Numbers related?
07:29 Proof Strategy
10:04 Change of Basis Matrices: An Example (LA51 Change of Basis Matrices)
15:32 Change of Basis Matrices: 3 important facts
18:47 Change of Basis Matrices for Polynomials: The setup (LA3 Polynomials, as an example of a v...)
21:14 Expanding falling factorials
21:41 Change of basis from falling factorials to powers
23:58 Powers in terms of falling factorials
24:30 Theorem: Change of Basis from powers to falling factorials
26:01 Strategy for Proof of Theorem
28:11 Proof of Theorem
31:34 Recap
32:18 Related Videos on Stirling Numbers

Videos on Stirling Numbers:
CO24: Stirling Numbers of the Second Kind CO24 The Stirling Numbers of the Seco...
CO25: A formula for Bell Numbers CO25 A formula for Bell Numbers
CO26: Formulas for Stirling Numbers of the Second Kind CO26 Formulas for Stirling Numbers of...
CO27: Stirling Numbers of the First Kind CO27 What are the Stirling Numbers of...
CO28: Stirling #s of 1st Kind & Falling Factorials CO28 Stirling #s of 1st Kind & Expans...
CO29: Stirling #s of 1st Kind vs 2nd Kind CO29 Stirling Numbers of First Kind v...

Part of a series of lectures on introductory Combinatorics. This full course is based on my book
Shahriar Shahriari, An Invitation to Combinatorics, Cambridge University Press, 2022.
DOI: https://doi.org/10.1017/9781108568708
For an annotated list of available videos see
https://pomona.box.com/s/by2ay2872avx...
YouTube Playlist: Combinatorics, An Invitation

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA, U.S.A.
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and five time winner of Pomona College's Wig teaching award.