World’s oldest Combinatoric Formula

‘yam?t?r?jabh?nasalagam’

Pronounced as yamaataaraajabhaanasalagam and written in sanskrit as ???????????????

This is a word used extensively throughout the ancient Sanskrit grammar works. The greatness of this word lies in the fact that

• This is the world’s oldest Combinatoric formula
• This is the world’s oldest known de Bruijn Sequence
• This is the world’s oldest known ‘Shift Register’
• This is one of the world’s oldest known memory wheel or mnemonic (because there are many other such memory wheels in ancient vedas)

First, let us see what this Sanskrit word represents. It represents a binary sequence as follows:

Consider the word ‘Canada’. This can be split into Ca-na-da where all three syllables require the same amount of time to pronounce.
Now consider the word ‘America’. This can be split into A-me-ri-ca where A-ri-ca require the same amount of time to pronounce where as ‘me’ in America requires twice the amount of time as ‘A’ or ‘ri’ or ‘ca’ to pronounce.

So in Sanskrit grammar we have short syllables (called Laghu) and long syllables (called Guru). Examples of laghu above are Ca,Na,Da in Canada and A,ri,ca in America. Example of Guru is the long syllable Me in America.

Short syllables are denoted by 1 and long syllables by 0. So we can write the syllables of the above mentioned sanskrit word ‘yamatarajabhanasalagam’ as 1000101110

Now let us see how this is a Combinatoric. Combinatorics is all about combinations. An example is, given 9 unique symbols, in how many unique ways can we pick a group of 3 symbols from it?

A second category of problems is a more tougher version of the above problem, given 3 unique symbols in how many unique ways can we pick a group of 9 symbols from it, by allowing repeated pick of a given symbol?

‘yamatarajabhanasalagam’ as a combinatoric represents a solution to one such problem of the second type where we have two symbols 0 and 1 and we have to find out as to how many unique groups of three can we pick up from it by allowing repeated picks?

The simple answer to the problem here is just divide the given formula into groups of three by shifting one place at a time as follows:

1000101110 = 100,000,001,010,101,011,111,110

Now this represents all the 8 possible combinations for the above mentioned problem of arranging 0 and 1 into groups of three! In other words, this is the list of all possible triplets of a binary sequence!

A Shift Register too!

Since the solution is in the form of shifting one place at a time from the left, this is also the world’s first ’shift register’! Note that shift registers are used extensively in modern computers to speed up calculations. Software programmers might be aware of left shifts and right shifts << and >>

Mnemonic

The main purpose of this word is to use it as a mnemonic or a memory wheel. That would make it easy for one to remember and quickly recall all possible combinations. There are hundreds of such words that are used in Sanskrit as mnemonics to help people memorize mathematical numbers and formulae. The authors of these mnemonics have been so creative that they have created sacred hymns, short stories etc which initially look like genuine hymns or short stories or sentences or proverbs or riddles etc, but when you decrypt them into numbers you end up with a hashing algorithm , or the value of PI to infinite decimal places, or with a logarithm etc, or with a formula etc…

The first memory wheel in modern history of mathematics appears only in 1882 where one such memory wheel was created by the French Mathematician Emile Baudot!! Can you imagine how advanced ancient Indian grammar and mathematics was!! Sometimes I feel we are actually living in a technologically inferior era compared to ancient Indians where today we are just reinventing the wheel !!

De Bruijn Sequence

These are special types of sequences first studied in modern history by de Bruijn and are defined as ‘Given a set S of words of length n, a de Bruijn sequence of span n is a periodic sequence such that every word in S (and no other n-tuple) occurs exactly once.’ In simple terms, de Bruijn sequences are nothing but the shift registers mentioned above!

Where is ‘yamAtArAjabhAnasalagam’ used?

In Sanskrit grammar this is used to divide poetry into a collection of three syllables called Ganas. ‘Yamatarajabhanasalagam’ defines all the 8 possible Ganas as follows

• ‘Ya’ Gana is 100
• ‘Ma’ Gana is 000
• ‘Ta’ Gana is 001
• ‘Ra’ Gana is 010
• ‘Ja’ Gana is 101
• ‘Ba’ Gana is 011
• ‘Na’ Gana is 111
• ‘Sa’ Gana is 110

The gana combinations are then used to define the rules for writing poetry. It is rules like these that form the basis of the most mathematical and scientific human spoken language – Sanskrit, which is why it was termed to be the only human spoken language with the ability to become a software programming language because of its precision – the research was done by the Forbes Magazine of Germany.

Which is why I have always felt that this language was not born on earth, but instead is of an alien origin, the language spoken by aliens with advanced technology and science. Those aliens were probably Type II or Type III civilizations!! Note that ancient vedic texts call Sanskrit as ‘Deva Bhaasha’ which means ‘A language of Divine (alien?) origin’

Well, divine Indeed !