Choosing From A Negative Number Of Things?? #SoME2
Channel: zhuli
Date: 2022-07-22

Combinatorial Reciprocity Theorems by Mattias Beck and Raman Sanyal: https://page.mi.fu-berlin.de/sanyal/teaching/crt/CRT-Book-Online.pdf

An introductory look at negative binomial coefficients, and in general, combinatorial reciprocity.

First, we explain how to formally justify negative binomial coefficients, then we explain what combinatorial reciprocity is, followed by an overview of some major examples and why it's important. Covered topics include multisets, the generalized binomial theorem, proper n-colorings, orientations (acyclic and totally cyclic), compatibility of n-colorings and orientations, nowhere-zero flows, posets (partially ordered sets), mappings of posets preserving order, lattice points in polygons (and polytopes in general). In the pinned comment, I reference the negative binomial distribution.

My Patreon: https://www.patreon.com/zhulimath/

00:00 Introduction
00:29 Chapter 1: Combinatorial Reciprocity
04:44 Chapter 2: Generating Functions
09:34 Chapter 3: More Examples of Combinatorial Reciprocity!
10:40 Chapter 3-1: Chromatic Polynomials
15:19 Chapter 3-2: Flow Polynomials
18:44 Chapter 3-3: Order Polynomials
21:48 Chapter 3-4: Ehrhart Polynomials
23:56 Conclusion

Intro riff taken from: Nikolai Kapustin - 8 Concert Etudes, Op. 40: III. Toccatina

Music Credit:
Dusty Decks / Fall Call
Timothy Infinite / Trumpet Man
Sarah, the Illstrumentalist / Just Maintaining
Gregory David / Corner Store
Chill Cole / Amber Lights
Guustavv / Coloration
Luwaks / Reversed Beginning
Giants' Nest / Tiny Things
Siine / Be Free With Me (instrumental)
courtesy of www.epidemicsound.com