I would like to find informations regarding this way of doing Integer Partitions or this conjecture,
Suppose you have all the ordered partitions of 5:
5
4 1
3 2
2 2 1
3 1 1
2 1 1 1
1 1 1 1 1
Then extend those partitions to 5 digits by adding 0:
5 0 0 0 0
4 1 0 0 0
3 2 0 0 0
2 2 1 0 0
3 1 1 0 0
2 1 1 1 0
1 1 1 1 1
And finaly for each extended partitions, count the digit repetitions, like this:
5 0 0 0 0 -> 1 4
4 1 0 0 0 -> 1 1 3
3 2 0 0 0 -> 1 1 3
2 2 1 0 0 -> 2 1 2
3 1 1 0 0 -> 1 2 2
2 1 1 1 0 -> 1 3 1
1 1 1 1 1 -> 5
And you can repeat !
5 0 0 0 0 -> 1 4 -> 1 4 0 0 0 -> 1 1 3 -> 1 1 3 0 0 -> 2 1 2 -> 2 1 2 0 0 -> 1 1 1 2 -> ..
4 1 0 0 0 -> 1 1 3 -> 1 1 3 0 0 -> 2 1 2 -> 2 1 2 0 0 -> 1 1 1 2 -> 1 1 1 2 0 -> 3 1 1
3 2 0 0 0 -> 1 1 3 -> 1 1 3 0 0 -> 2 1 2 -> 2 1 2 0 0 -> 1 1 1 2 -> 1 1 1 2 0 -> 3 1 1
2 2 1 0 0 -> 2 1 2 -> 2 1 2 0 0 -> 1 1 1 2 -> 1 1 1 2 0 -> 3 1 1 -> 3 1 1 0 0 -> 1 2 2
3 1 1 0 0 -> 1 2 2 -> 1 2 2 0 0 -> 1 2 2 -> 1 2 2 0 0
2 1 1 1 0 -> 1 3 1 -> 1 3 1 0 0 -> 1 1 1 2 -> .... -> 1 2 2
1 1 1 1 1 -> 5 -> 5 0 0 0 0 -> 1 4 -> 1 4 0 0 0 -> 1 1 3 -> ... -> 1 2 2
All of them finish in 1 2 2!
I want informations, documentations
Thanks a lot!