Unrestricted Partitions

Some may have noticed that I have been giving a representation of the countdown clock.

The first line is the fundamental theorem of arithmetic.  That is, every integer $$n$$ can be represented by a unique factorisation of prime numbers.  It is a multiplicative function.

The second line is an additive function.

Unrestricted partitions is one of the most fundamental problems in additive number theory.

The unrestricted partition function,$$p(n)$$, is the number of ways that $$n$$ can be written as a sum of positive integers $$\le n$$.

The number of summands (or addends or parts) is unrestricted, repetition is allowed and the order of the summands is not taken into account.

For example :

$$p(5) = 7$$

That is :

$$5 = 4 + 1 = 3 +2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1$$

The values for $$p(n)$$ grow very rapidly :

$$p(10) = 42\\ p(100) = 190,569,292\\ p(200) = 3,972,999,029,388$$

Ramanujan gave us this estimate :

$$\displaystyle p(n) \sim \frac {e^{\pi \sqrt{2n/3}}}{4n \sqrt 3}$$

We have already considered a restricted partition function in Euler’s Pentagonal Number Theorem where we investigated the partitions with distinct (unequal) summands.  There is another nice result regarding distinct summands.

We will consider geometric representations, generating functions, a recursion formula by Euler and other things including the Goldbach Conjecture which is a particular partition function.

© OldTrout $$2019$$

No Audio file – Does not translate well