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Mathematics is also potent medicine.
It remains to this day the most frequently prescribed medicine for …
Counting
The most primordial of mathematical operations
And sometimes the most intricate
Triangular Numbers
Square Numbers
Wallace and Gromit: A Close Shave (1995)
Aardman Animations
Some beautiful counting problems
not involving sheep
Act I
Kayaking on the Adyar
The kayakers
Solo or Tandem
🛶 The Adyar River
A group of friends wants to go kayaking on the Adyar river.
How many different boating configurations are possible?
2 Friends 2 configurations
1 all solos · 1 as a pair
3 Friends 4 configurations
1 all solos · 3 with one pair
4 Friends 10 configurations
1 all solos · 6 with one pair · 3 with two pairs
9 friends
The encoding
The code
reading the configuration
The code words
Read columns: 1↔3 · 2↔4
Kayak configuration
Read columns: 1↔4 · 2 solo · 3 solo
Kayak configuration
5 Friends 26 configurations
1 all solos · 10 with one pair · 15 with two pairs
🛶 Counting the configurations
How many configurations if there are n friends?
Spot the pattern ?
The pattern
Act II
Madras Port Trust
Configurations of boxes
📦 The Loading Dock
Containers need to be loaded onto a ship.
How many different stacking configurations are possible?
Stacking 3 boxes ·1 / 4
The stacking rule
Left justified configurations: No gaps allowed
Case A
Case B
Stacking 4 boxes ·1 / 10
Stacking 5 boxes ·1 / 26
📦 Counting the Configurations
How many ways to stack n boxes?
n = 1
1
n = 2
2
n = 3
4
n = 4
10
n = 5
26
n = 6
?
n = 7
?
…
Coincidence ? Or something deeper ?
Interlude
The Chennai Port is fertile breeding ground not just for mathematical problems, but also for …
Srinivasa Ramanujan
1887 – 1920
All configurations · n = 5
26 configurations
Partitions
n = 1
1
n = 2
2
1+1
n = 3
3
2+1
1+1+1
n = 4
4
3+1
2+2
2+1+1
1+1+1+1
n = 5
5
4+1
3+2
3+1+1
2+2+1
2+1+1+1
1+1+1+1+1
n = 6
6
5+1
4+2
4+1+1
3+3
3+2+1
3+1+1+1
2+2+2
2+2+1+1
2+1+1+1+1
1+1+1+1+1+1
Two stories, one pattern
🛶kayaks
1
2
4
10
26
76
232
…
📦stacking
1
2
4
10
26
76
232
…
Kayak arrangements & stacking configurations: both counts are equal.
Act III
Why do these very different counting problems have the same solution ?
Kayak (code): numbers
Stack of boxes: shape
They share the same mathematical DNA
They just look different, that's all.
✨
Can we magically transform a kayak configuration into a stack of boxes?
✨
Can we extract a shape from a list of numbers (code)?
Yes !
Shapes from numbers: Ups and Downs and in-betweens
6 2 1 8 3 9 2 4 5 4 8 3
The song of Aadhaar
The RSK Algorithm
The Robinson–Schensted–Knuth Algorithm
List of numbers → Stack of boxes
Robinson · Schensted · Knuth
G. de B. Robinson
1906 – 1992
Craige Schensted
1927 – 2021
Donald E. Knuth
b. 1938
The one-way Bharat Express
1
Single file seating — one row of seats, passengers seated one behind the other.
2
No seat numbers — passengers sit in height order (for unobstructed TV view).
3
Entry and exit corridors are one-way — no turning back.
RSK · Step by Step
Filling the Trains
RSK · Auto
Filling the Trains
RSK · Auto
Filling the Trains
RSK · Result
Filling the Trains
Homework
Apply the RSK algorithm to your Aadhaar number and see what shape you get.
There are only 75 different shapes possible !
Top 3 most common:
Rarest of rare:
The Kayaks and Boxes mystery solved
RSK transforms one into the other
Kayak
The mathematics that underlies …
The theory of symmetry
Group Representations
The mathematics that underlies …
Communication
Particle Physics
James Webb Telescope
Group Representations: a far-reaching theory of symmetries
Some everyday applications are:
Processing digital voice signals on cell phone networks
Behaviour of quarks and fundamental particles
Molecular spectroscopy and detecting elements in stars and exoplanets
Stacks of numbered boxes (tableaux) are related to group representations
Particle Physics
The Eightfold Way
Representation Theory
Lakshmibai–Seshadri: reimagined the mathematical framework from the ground up
V. Lakshmibai
1944 – 2023
C. S. Seshadri
1932 – 2020
Lakshmibai–Seshadri paths: a grand unification
Young diagram
27 tableaux
Prior to this, mathematicians had piecemeal tools for different kinds of symmetry.
Lakshmibai and Seshadri built a single unified framework that works for all of them
— finite or infinite, simple or exotic — in one stroke.
Prof. Seshadri lecturing on the path model at CMI's original location in T. Nagar, early 2000s. These paths are now known as Lakshmibai–Seshadri paths.
