8 Jan
2016
8 Jan
'16
10:48 p.m.
Aha. Very interesting. —Dan
On Jan 8, 2016, at 9:41 PM, James Propp <jamespropp@gmail.com> wrote:
The general problem is to find n+1 permutations of the sequence 1,2,3,...,2n such that all (n+1)(2n-1) partial sums are distinct, or equivalently, such that every integer strictly between 0 and 1+2+3+...+2n occurs exactly once as a partial sum. (Here a partial sum of a sequence is neither allowed to be empty nor allowed to coincide with the original sequence.)
E.g., for n=2, we can use 1,2,3,4 (with partial sums 1,3,6) and 4,3,1,2 (with partial sums 4,7,8) and 2,3,4,1 (with partial sums 2,5,9).
It seems that this can be done for all n, but nobody knows how to prove it.