More sequences · [Nov 14, 12:26 PM]
Suppose we have a set of 4 vectors: $$\{ (1,-1,-1) , (-1,1,-1),(-1,-1,1),(1,1,1)\},$$ and we want to make a sequence $\{R_i\}_{i=1}^N$ of these vectors with a zero sum. You could easily convince yourself that you need to use all of them at least once with $N=4$.
Now suppose instead of the sum you want to cancel the following for all $\alpha\ne\beta$: $$\sum_{i<j} R_i^\alpha R_j^\beta $$ where $R_i^\alpha$ refers to the coordinate $\alpha$ of the $i$-th vector. Smallest such sequence is 8 steps long. It’s easy to show that the length of such sequence has to be a multiple of 4 and 4 itself doesn’t work but 8 does.
Now how could I do the same thing with a growing monster like this:
$$\sum_{i<j<k} R_i^\alpha R_j^\beta R_k^\gamma$$
Comment
Commenting is closed for this article.