Signpost sequence

Mathematically, a signpost sequence is a localized sequence, meaning the ${\displaystyle n}$th signpost lies in the ${\displaystyle n}$th interval with integer endpoints: ${\displaystyle s_{n}\in (n,n+1]}$ for all ${\displaystyle n}$. This allows us to define a general rounding function using the floor function:

${\displaystyle \operatorname {round} (x)={\begin{cases}\lfloor x\rfloor &x<s(\lfloor x\rfloor )\\\lfloor x\rfloor +1&x>s(\lfloor x\rfloor )\end{cases}}}$

Where exact equality can be handled with any tie-breaking rule, most often by rounding to the nearest even.

In the context of apportionment theory, signpost sequences are used in defining highest averages methods, a set of algorithms designed to achieve equal representation between different groups.^[2]