The Art of Bijective Combinatorics Part I
An introduction to enumerative, algebraic and bijective combinatorics
The Institute of Mathematical Sciences, Chennai, India (January-March 2016)
Ch 3 Exponential structures and generating functions
Ch 3a
9 February 2016
slides_Ch3a (pdf 12 Mo) video Ch 3a (1h 17mn)
species and structures 3 00’ 35’’ video
generating function of a species 9 8‘ 59’‘ video
examples of species 10 11‘ 56’‘ video
complements: formalization of species 19 28’ 26’’ video
operations of species 22 30’ 32’’ video
sum 23 30’ 35’’ video
product 24 32’ 41’’ video
example: derangements 25 35‘ 59’’ video
substitution 28 40’ 35’’ video
F-assemblée of G-structures 29 42‘ 50’‘ video
example of «assemblées» 31 46‘ 01’‘ video
endofunctions as substitutions of arborescences in permutations 36 50‘ 48’‘ video
pointed species 43 53’ 34’’ video
vertébrés 46 57‘ 37’‘ video
derivative 49 1h 00’ 59’’ video
a typical «species proof» 52 1h 7‘ 9’‘ video
the end 57 1h 17’ 8’’
Ch 3b
11 February 2016
slides_Ch3b (pdf 12 Mo) video Ch 3b (1h 19)
weighted species 3 0’ 20’’ video
exercise: «assemblée» of permutations and Lah numbers 6 2‘ 48’‘ video
weighted species: definition 7 3’ 30’’ video
generating function for weighted species 8 4‘ 56’‘ video
operations of weighted species 9 5’ 38’’ video
examples: some orthogonal polynomials 14 11’ 42’’ video
Hermite polynomials 15 11’ 48’’ video
Laguerre polynomials 20 15’ 41’‘ video
bijective proof of Mehler identity for Hermite polynomials 26 21’ 56’’ video
Sheffer polynomials 35 25’ 37’’ video
Stirling numbers 1st kind 38 29’ 33’’ video
Stirling numbers 2nd kind 39 30’ 54’’ video
Linear species (or L-species) 41 35’ 22’’ video
example of L-species: increasing binary trees 44 39’ 16’’ video
derivative of an L-species 56 43’ 10’’ video
integral of an L-species 61 48’ 47’’ video
some historical remarks about tangent and secant numbers 70 1h 1’ 13’’ video
the end 80 1h 5’ 37’’ video
complements to Ch 3 1h 5’ 37’‘ video_Ch3b-complements slides_Ch3b-complements
complement 1: combinatorial methods in control theory 2 1h 5’ 37’‘ video
iterated integral 5 1h 7’ 26’‘ video
shuffle product 7 1h 8‘ 45’‘ video
an example 8 1h 10’ 36’‘ video
combinatorial resolution (of a differential equation with forced term) 12
complement 2: combinatorial solution of differential equations with species 21 1h 15’ 10’‘ video
separation of variables 24 1h 15’ 38’‘ video
separation of variables: extensions and iterated integrals 29 1h 16’ 43’’ video
complement 3: elliptic and Dixon functions, Polya urn model 35 1h 17’ 22’’ video
complement 4: (formal) orthogonal polynomials 39 1h 18’ 02’’ video
the end 46 1h 19’ 26’’