Media Summary: We prove several partition identities, including Euler's pentagonal number theorem. We prove that the number of k-dimensional subspaces of a finite vector space F_q^n is a q-binomial coefficients. We discuss ... We study the Stirling numbers of the second kind. Then we discuss 12 variants of the problem: How many ways are there to put n ...
Lecture 9 Enumerative Combinatorics Federico - Detailed Analysis & Overview
We prove several partition identities, including Euler's pentagonal number theorem. We prove that the number of k-dimensional subspaces of a finite vector space F_q^n is a q-binomial coefficients. We discuss ... We study the Stirling numbers of the second kind. Then we discuss 12 variants of the problem: How many ways are there to put n ... We complete the proof of Schreyer's theorem, and begin to discuss free resolutions of modules. What do the most common operations on power series correspond to combinatorially? We introduce the "Symbolic Method" that ... We show two ways of representing a permutation as a tree. We define q-binomial coefficients and give a
We discuss the Möbius function and Möbius inversion for Boolean posets, divisor posets, distributive lattices, and partition lattices. We introduce the compositional formula for exponential generating functions and illustrate it by counting ordered set partitions. We count labeled trees and parking functions. We discuss the "Symbolic Method" for *labeled* combintaorial objects and their *exponential*. We use it to revisit and better ... We prove the formula for Catalan numbers, and show that the number of 321-avoiding permutations is given by a Catalan number ... We count the rhombus tilings of a hexagon. We interpret the determinants of the Catalan numbers and Schröder numbers ...
