Stirling numbers and Pascal triangles | Geometric Linear Algebra 23 | NJ Wildberger

When we interpret polynomials as sequences rather than as functions, new bases become important. The falling and rising powers play an important role in analysing general sequences through forward and backward difference operators.

The change from rising powers to ordinary powers, and from ordinary powers to falling powers give rise to two interesting families of numbers, called Stirling numbers of the first and second kind. We use Karamata notation, advocated by Knuth to describe these: brackets and braces. Combinatorial and number theoretic interpretations are mentioned.

We discuss the important relation between two bases of a a linear space and the corresponding change from one kind of coordinate vector to another. This is applied to study general polynomial sequences.

This lecture is not easy, and represents a high point of this course in Linear Algebra. However it introduces powerful and common techniques which are actually quite useful in a variety of practical applications.

CONTENT SUMMARY: pg 1: @00:08 Intro: (Stirling numbers and Pascal triangles); sequences; change of terminology @00:44 ; falling power; rising power; list of rising powers; summation notation and Stirling numbers @03:00;
pg 2: @04:55 James Stirling (1749), "Methodus Differentialis"; Stirling number notation warning @05:04 ; 'n bracket k' as Karamata notation (Knuth); Stirling numbers of the first kind; Change of basis rewritten from pg 1 @05:29 ; Stirling matrix of the first kind; remark about unconventional indexing of Stirling numbers @06:36;
pg 3: @07:13 Calculating Stirling numbers; Theorem (Recurrence relation: Stirling numbers); proof;
pg 4: @10:56 Pascal's triangle and binomial coefficients; recurrence relation for binomial coefficients; Pascal matrix;
pg 5: @14:08 Combinatorial interpretation of Sterling numbers;
pg 6: @17:34 Number theoretic interpretation of Sterling numbers; summary of Sterling number interpretation @21:50;
pg 7: @23:09 Sterling numbers of the 2nd kind; Inverting the Pascal matrices;
pg 8: @26:36 Inverting Stirling matrices; reintroduction of some ignored symmetry @27:48 ; Sterling matrix of the 2nd kind;
pg 9: @30:41 Definition of Stirling numbers of the second kind; 'n brace k' notation of Stirling numbers of the 2nd kind; Sterling matrix of the 2nd kind;
pg 10: @32:54 Combinatorial interpretation of Sterling_numbers_2nd_kind ; Theorem (Recurrence relation for Sterling_numbers_2nd_kind);
pg 11: @35:54 Statement of the importance of the Sterling numbers; important question @37:23 ; suggestion to review starting WLA1_pg7 @40:27;
pg 12: @40:48 Of primary importance to problems of practical application; Non_standard ideas; This is at the heart of change of basis @47:08;
pg 13: @47:26 Transpose a matrix and vector;
pg 14: @50:11 Application of this (effect of change of basis on coordinate vectors): analyse a polynomial sequence; Newtons formula; A very useful thing to be able to do @53:54;
pg 15: @54:44 General C: transpose of signed Stirling matrix of 1st kind;
pg 16: @55:30 Exercises 23.1-3;
pg 17: @56:13 Exercises 23.4-5; closing remarks @57:14; (THANKS to EmptySpaceEnterprise)

Video Chapters:
00:00 Introduction
4:56 James Stirling Methodus Differentialis
10:56 Pascal Matrix
14:08 Combinatorial interpretation
17:34 Number theoretic interpretation
23:08 Inverting Pascal matrices
26:36 Inverting Stirling matrices
30:40 Stirling numbers of the second kind
32:54 Combinatorial interpretation of Stirling numbers
50:11 Square pyramidal numbers
54:44 Transpose of signed Stirling matrix of first kind
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