Signpost sequence

{{Short description|Generalized rounding rule}} In mathematics and apportionment theory, a '''signpost sequence''' is a sequence of real numbers, called '''signposts''', used in defining generalized rounding rules. A signpost sequence defines a set of ''signposts'' that mark the boundaries between neighboring whole numbers: a real number less than the signpost is rounded down, while numbers greater than the signpost are rounded up.'''<ref name=":4">{{Citation |last=Pukelsheim |first=Friedrich |title=From Reals to Integers: Rounding Functions, Rounding Rules |date=2017 |work=Proportional Representation: Apportionment Methods and Their Applications |pages=71–93 |url=https://doi.org/10.1007/978-3-319-64707-4_4 |access-date=2021-09-01 |place= |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-64707-4_4 |isbn=978-3-319-64707-4|url-access=subscription }}</ref>'''

Signposts allow for a more general concept of rounding than the usual one. For example, the signposts of the rounding rule "always round down" (truncation) are given by the signpost sequence <math>s_0 = 1, s_1 = 2, s_2 = 3 \dots</math>

== Formal definition == Mathematically, a signpost sequence is a ''localized'' sequence'','' meaning the <math>n</math>th signpost lies in the <math>n</math>th interval with integer endpoints: <math>s_n \in (n, n+1] </math> for all <math>n </math>. This allows us to define a general rounding function using the floor function:

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

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

== Applications == 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.<ref name=":02">{{cite book |last1=Balinski |first1=Michel L. |url=https://archive.org/details/fairrepresentati00bali |title=Fair Representation: Meeting the Ideal of One Man, One Vote |last2=Young |first2=H. Peyton |publisher=Yale University Press |year=1982 |isbn=0-300-02724-9 |location=New Haven |url-access=registration}}</ref>

== References == {{Reflist}}

Category:Sequences and series Category:Apportionment methods