Differents ways to represent a natural number

Asked Jul 12 '21 at 17:53

Active Jul 13 '21 at 19:13

Viewed 45 times

I am lookin for the differents ways to represent a natural number just like in this example: the number 4 can be expressed in these following 8 ways:

1+1+1+1
2+1+1
1+2+1
1+1+2
3+1
1+3
2+2
4

I have been able to verify that the representation up to n={1,2,3,4,5} corresponds to $2^{n-1}$ but i wish i could prove it for n

edited Jul 13 '21 at 19:13

jake_1212

asked Jul 12 '21 at 17:53

Mogul

4

These are called compositions and there are $2^{n-1}$ of them (see the link). – lulu Jul 12 '21 at 17:55
You may also be interested in Young tableaux https://en.wikipedia.org/wiki/Young_tableau – Tristan Duquesne Jul 12 '21 at 17:57
Thank you @lulu thats was very useful! – Mogul Jul 12 '21 at 18:00
I would be inclined to include "iv" as a way to indicate "4" as well as "$\cdot\cdot\cdot\cdot$". – user247327 Jul 13 '21 at 19:21

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