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 Ch3a  link to YouTube    (1h 17mn)
video Ch3a  link to bilibili
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 Ch3b  link to YouTube     (1h 19')
video Ch3b  link to bilibili
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-complementsslides_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’’