CoursE  X.V.  IMSc  2016

 
 

Course  IMSc  Chennai, India 

January-March 2016

An introduction to enumerative, algebraic and

bijective combinatorics



Ch 3   Exponential structures and generating functions

          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’’


         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                             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’’



last update: 16  April 2016



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