Editing Combinatorics (section)

== Definition ==
The full scope of combinatorics is not universally agreed upon.<ref>{{cite web |last=Pak |first=Igor |title=What is Combinatorics? |url=https://www.math.ucla.edu/~pak/hidden/papers/Quotes/Combinatorics-quotes.htm |access-date=1 November 2017 |archive-date=17 October 2017 |archive-url=https://web.archive.org/web/20171017075155/http://www.math.ucla.edu/~pak/hidden/papers/Quotes/Combinatorics-quotes.htm |url-status=live }}</ref> According to [[H. J. Ryser]], a definition of the subject is difficult because it crosses so many mathematical subdivisions.<ref>{{harvnb|Ryser|1963|loc=p. 2}}</ref> Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with:
* the ''enumeration'' (counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems,
* the ''existence'' of such structures that satisfy certain given criteria,
* the ''construction'' of these structures, perhaps in many ways, and
* ''optimization'': finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other ''optimality criterion''.

[[Leon Mirsky]] has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained."<ref>{{citation |last=Mirsky |first=Leon |title=Book Review |url=https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979-14606-8/S0273-0979-1979-14606-8.pdf |journal=Bulletin of the American Mathematical Society |volume=1 |pages=380–388 |year=1979 |series=New Series |doi=10.1090/S0273-0979-1979-14606-8 |doi-access=free |access-date=2021-02-04 |archive-date=2021-02-26 |archive-url=https://web.archive.org/web/20210226080424/https://www.ams.org/journals/bull/1979-01-02/S0273-0979-1979-14606-8/S0273-0979-1979-14606-8.pdf |url-status=live }}</ref> One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella.<ref>{{cite book |last1=Rota |first1=Gian Carlo |url=https://link.springer.com/book/10.1007/978-0-8176-4775-9 |title=Discrete Thoughts |date=1969 |publisher=Birkhaüser |isbn=978-0-8176-4775-9 |page=50 |doi=10.1007/978-0-8176-4775-9 |quote=... combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent ... . The typical ... case of this is algebraic topology (formerly known as combinatorial topology)}}</ref> Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, [[Countable set|countable]]) but [[Discrete mathematics|discrete]] setting.