Consider a bijection $f: \mathbb{N} \to \mathbb{N}$:
We define the (possible infinite) number of "crosses" or "intersections" by the following:
For every pair $i,j$ with $i<j, $ define:
$$ \delta_{ij}=\begin{cases} 0&\ \text{if} \ f(j) > f(i)\\[8pt] 1&\ \text{if} \ f(j) < f(i)\\[8pt] \end{cases} $$
Then the number of (possibly infinite) intersections is $\ s(f) = \displaystyle \sum_{i=1}^\infty \left( \sum_{j=i+1}^\infty\delta_{ij} \right)$.
(s depends only on the bijection $f$).
Also, define a bijection eventually equal to the identity to be one in which $ \exists N$ such that $f(n) = n \quad \forall n \geq N$.
My conjecture is that a bijection $f: \mathbb{N} \to \mathbb{N}$ is eventually equal to the identity $ \iff s$ is finite, but I don't know how to prove this formally.
Note that my conjecture would be false with bijections $f: \mathbb{Z} \to \mathbb{Z}$, for example:
The pigeonhole principle came to mind, but I'm not sure if it's useful or necessary for a proof.
from Hot Weekly Questions - Mathematics Stack Exchange
Adam Rubinson
Post a Comment