CoursE  X.V.  IMSc  2016

 
 

Course  IMSc  Chennai, India 

January-March 2016

An introduction to enumerative, algebraic and

bijective combinatorics



Ch 5   Tilings, determinants and non-intersecting paths


                  1 March  2016

                           slides_Ch5a       (pdf,   25 Mo)                    video    (1h 14mn)


The  LGV  Lemma    slide  3   video  15’’  video

        the crossing condition  (C)  5’ 23’’  video

a  simple  example  12    8’ 59’’  video

proof of the LGV Lemma  16    11’ 38’’  video

        a Lemma from « the Book »  24  17’ 35’’  video

        why LGV ?  27  20’ 02’‘   video

binomial  determinants  29  21’  40’‘   video

        main proposition  36  23’ 36’‘    video

        3 corollaries  38, 42  25’  12’‘   video

example: Narayana numbers and number of Baxter permutations  43  28’  43’’  video

formulae for binomial determinant  49  30’  19’’  video

        first formula with contents and hook-lengths  54  34’  20’‘   video

        contents 60  39’  28’’  video

        an example  63  42’  44’’  video

second formula for binomial determinant  (exercise)  65  45’  09’’  video

semi-standard Young tableaux  68  46’  40’’  video

        formula for the number of semi-standard Young tableaux  76  54’  03’’  video

        example  77  54’  45’’  video

formulae for Narayana numbers and for the number of Baxter permutations  80  56’  05’’  video

binomial determinants: other example with permutations having a given up-down sequence  87  1h  0’  14’’  video

plane partitions  92  1h  4’  13’’  video

        MacMahon formula for plane partitions in a box  96  1h  7’  33’’  video

paths for plane partition  97  1h  8’  38’’  video

        proof of MacMahon formula  104  1h  10’  06’’  video

the end  111  1h  14‘  05’’ 



               3 March 2016

                                 slides_Ch5b           (pdf,  35 Mo)         video  (1h 35mn)


from the previous lecture      slide 3  0’  26’‘   video

Jacobi identities for Schur functions  12  4’  03’’  video

        homogeneous symmetric functions  14  5’  13’’  video

        elementary symmetric functions  15  6’  50’’  video

Jacobi identities for skew Schur functions  23  23’  10’’  video

Duality  (the idea of duality in paths)  32  26’  27’’  video

        dual configurations of paths  37  38’  26’’  video

        inverse of the Fermat matrix of binomial coefficients  38  28’  34’’  video

TASEP and MacMahon-Narayana-Kreweras determinant  42  29’  58’’  video

        stationary probabilities for the TASEP with pair of Ferrers diagram  43  30’ 09’’  video

        proof of the determinant formula with dual configuration of paths  45  31’  58’’  video

        (alpha,beta)-analog of the MacMahon-Narayana-Kreweras determinant  48  35’  55’’  video

Kreweras determinant for the enumeration of chains in the Young lattice  49  36’  38’‘   video

Orthogonal polynomials: expressions for the coefficient b_k and lambda_k

                          with Hankel determinants of moments  51  38’  27’’  video

        Hankel detreminant: definition  55  41’  10’’  video

        Hankel determinant and configuration of non-intersecting Dyck paths  56  41’  34’’  video

        virtual crossing for configuration of non-intersecting Motzkin paths  57-58  44’  44’’  video

        expression for lambda_k  61  46’  43’’  video

        expression for b_k  62  47’  59’’  video

Tilings  63  50’ 00’’  video

        formula for the number of tilings of a m x n rectangle  66  50’  52’’  video

Tilings on triangular lattice  69  52’  49’’  video

        exercises  72, 73  53’  55’’  video

        tilings and plane partitions 75-76  54’  38’’  video

Tilings and perfect matchings  79  55’  32’’  video

Aztec tilings  87  57’  20’’  video

Bijection Aztec tilings -- non-intersecting Schröder paths  94  58’  57’’ video

«Bijective computation» of the Hankel determinant of Schröder numbers  103  1h  6’  07’’  video

        from Schröder paths to weighted Dyck paths  108  1h  7’  00’’  video

        from Ch 2: two different interpretation of the (beta)-distribution for Dyck paths  109  1h  7’  23’’  video

Complements: Schröder numbers and the associahedron  114  1h  10’  39’’  video

        Schröder tree  119  slide missing in the video

        exercise: Schröder trees and Schröder paths  120  slide missing in the video

        Hipparchus number  122  1h  14’  40’‘   video

Complements: back to the formula for the numbers of tilings of a rectangle  125  1h  17’  36’’  video

The arctic circle theorem  130  1h  20’  00’’  video

Perfect matchings and the Pfaffian methodology  142  1h  25’  47’’  video

        Pfaffian  145  1h  26’  42’’  video

        admissible orientations and Kastelyn theorem  148-149  1h  28’  06’’  video

        Bijections for the Ising model  150-151  1h  29’  39’’  video

In conclusion: a nice formula 152  1h  31’  12’’  video

        Hankel determinant of Catalan numbers  153  1h  31’  21’’  video

A festival of bijections  155  1h  31’  51’’  video

the end  165  1h  35’  23’’

       

       



last update: 16 April 2016


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