First part of the talk 

 

Presentation by Aniruddhan Ganesaraman (CMI)     0:29

Trees in Nature   2     2:32   

From trees in Nature to mathematical trees   12     3:34

 trees everywhere in computer science with D; Knuth  20

Binary tree   21     4:26

definition

enumeration of binary trees: 1, 2, 5, 14

exercise 1   27     6:04

Some sequences ...  30     7:54

Fibonacci numbers, n! and permutations, prime numbers sequence

experimental combinatorics, guess a formula  35

the on-line encyclopedia of integer sequenes (oeis.org)   38

formula for Catalan numbers, Eugene Catalan   40

The arithmetical triangle (binomial coefficients)   43     13:53

"Pascal triangle", "Yang Hui triangle", Pingala in ancient  India

Bijective combinatorics, Catalan numbers   51     17:58

Injections, surjections, bijections ...   59     19:21

definitions, one-to-one correspondence, reverse bijection

Permutations   70     22:04

sub-exedante function, bijection with permutations

exercise 2   73     25:06

Binomial coefficient   76     25:33

proof of the formula for binomial coefficient

Increasing binary trees   80     27:06

exercise 3   83     27:43

Binomials coefficiens with paths   84     28:13

A bijective proof with binomial coefficients   91     32:04

exercise 4   94

Another bijective proof with binomial coefficients   95     33:57

matchings, Fibonacci numbers and binomials coefficients

exercise 5  a)  98     35:37   b)   99     36:01  

Pingala and meters in Sanskrit

At the primary school .... additions, subtractions, divisions   101     37:26

playing with the arithmetical triangle

exercise 6   109     40:56

At the primary school: additions   109     41:20

"the Catalan triangle"

Binary trees and Dyck paths   114

the bijection binary trees --- Dyck paths (with violins)  118     43:34

exercise 7   121     45:06

At the prilmary school: subtractions, the reflexion principle   122     45:35

exercise 8 "ballot numbers"  128-129     49:08

Arbogast triangle (1800)   130

At the primary school: divisions   131     51:08

execise 9   multiplicative recurrence relation for Catalan numbers 135   52:19

Bijective proof for the multiplicative recurrence relation of Catalan numbers   136

exercise 10   144     55:30

images of random binary tree and increasing binary tree  145    56:25

 

Second part of the talk

 

Analytic proof   2     58:37

Formal power series   6     1:01:57

Analytic proof with formal power series   18     1:04:46

Modern combinatorics: operations on combinatorial objects   23     1:07:49

The bijective paradigm   28     1:10:00

Lagrange and figures   

"replacing calculus by figures and bijections"

an identity with Ramanujan continued fraction   32   1:11:25

bijection Dyck paths --- heaps of dimers (pyramids)  (with violin)  36  1:13:24

conversely, "making calculus from the visual figures"   38

Feynman diagrams and binary trees   42    1:14:56

Triangulations of a convex polygon   45     1:15:30

A letter from Leonhard Euler to Christian Goldbach (1751)   49     1:15:59

bijection triangulations --- binary trees (with violins)  54     1:17:13

exercise 11 reverse bijection   61     1:18:44

Relation with physics of dynamical systems: the TASEP   62     1:18:48

execise 12   75     1:22:14

A final surprise ...

molecular biology, computer science, hydrogeology  76     1:22:34

molecular biology: trees in RNA secondary structures        

forest of planar trees

complexity of the forest of planar trees underlhying the RNA molecule

minimum number of registers needed to commpute an arithmetical 

expression

hydrogeology: Strahler number of a binary tree

3 identical generating functions, relation with the arithmetical triangle and Fibonacci numbers

Thank you !   94     1:27:44

Saraswati, the godess of knowledge, art and science   95     1:28:17

some questions     1:28:34

the end     1:35:23