One can regard the Young diagram as a visual notation for the partition. 'Introduction to Combinatorics' by Slomson has a good elementary chapter on partititions.

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5^3 means 125 not 3 fives.

Juan Weisz In combinatorics a superscript is often used to indicate number of repetitions in a list, for example when writing down the spectrum of a graph. It depends on the context.

Repetitions are not partitions.

Partitions:

15=1+14=2+13=3+12 etc

A partition does allow repetitions. However partitions with unequal parts are also of interest. It is known, for example, that the number of partitions into unequal parts is the same as the number of partitions into odd parts.

Juan Weisz I invite you to see the attached file to understand!

Ok, fair enough.

For 'parts' you can read 'summands'. So for example 8 = 1+2+5 is a partition of 8 with unequal parts. And 1+1+1+5 is a partition of 8 with all odd parts. The nice fact is that the number of such partitions is the same :D

@Juan Weisz Perhaps 'partition' is used differently in a physics context? :)

That might work. The approach that I know uses generating functions. The generating function for the number of 'unequal' partitions is (1+x)(1+x^2)(1+x^3)(1+x^4)...., whereas the generating function for partitions into odd parts is the product of all terms (1+x^t+x^{2t}+x^{3t}+...) where t is odd. Then we just need to show that the resulting functions are the same. This explained very well in the book by Slomson that I mentioned in the beginning.

In physics counting problems arise in satistical mechanics

..how many microstates are consistant with a macrostate.

As in the ising model, not solved yet for all dimensions.

Then the counting is different depending on

The kind of statistics, classical

Or 2 kinds of quantum ones.

Would urge pure mathematicians to

Learn some of this.eg book by Pathria.

Not too hard if you start off with ideal

Gases.

Tolmans old book also impresive.

Thank you Juan Weisz . Peter Breuer In this case the products will converge for 0 < x < 1 :)

For any positive integer t and 0 < x < 1, we have 1+x^t+x^{2t}+... = 1/(1-x^t) < 1+((x^t)/(1-x)) < e^((x^t)/(1-x)). The generating function is thus bounded above by e^(x+x^2+x^3+...)/(1-x) < e^(x/(1-x)^2). Not my own argument, just quoting Slomson :)