§ Catalan numbers as popular candidate votes (TODO)
- Usually, folks define catalan numbers as paths that go up or right from to in a way that never goes below the line .
I quite like the latter interpretation, because we really are counting
two different things (votes for and ) and then expressing a relationship
between them. It also allows us to directly prove that
- The catalan numbers can be thought to model two candidates and such that during voting, the votes for never dip below the votes for .
catalan(n) is equal
to by reasoning about seqences of votes, called as
§ Ballot sequences