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 5   Tilings, determinants and non-intersecting paths
Ch 5a

1 March  2016
slides_Ch5a       (pdf,   25 Mo)                  
video Ch5a  link to YouTube   (1h 14mn)
video Ch5a  link to bilibili
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’’ 

Ch 5b

3 March 2016
slides_Ch5b           (pdf,  35 Mo)        
video Ch5b  link to YouTube  (1h 35mn)
video Ch5b  link to bilibili
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’ 52’’
the end  165  1h  35’  23’’