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 (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’’
Ch 5b
3 March 2016
slides_Ch5b (pdf, 35 Mo) video_Ch5b (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’ 52’’
the end 165 1h 35’ 23’’