This question might seem strange, but I had the feeling it's possible to decompose in a unique way a number as follows:
if $x < n!$, then there is a unique way to write x as: $x = a_1\cdot 1! + a_2\cdot 2! + a_3\cdot3! + ... + a_{n-1}\cdot(n-1)!$ where $a_i \leq i$
I looked at factorial decomposition on google but I cannot find any name for such a decomposition.
example:
If I chose :
(a1,a2) =
- 1,0 -> 1
- 0,1 -> 2
- 1,1 -> 3
- 0,2 -> 4
- 1,2 -> 5
I get all number from $1$ to $3!-1$
ideas for a proof:
The number of elements between $1$ and $N!-1$ is equal to $N!-1$ and I have the feeling they are all different, so this decomposition should be right. But I didn't prove it properly.
Are there proofs of this decomposition? Does this decomposition as a name? And above all is this true ?
Thanks in advance