• Usually, folks define catalan numbers as paths that go up or right from $(1, 1)$to $(n, n)$ in a way that never goes below the line $y = x$.

• The catalan numbers can be thought to model two candidates $A$ and $B$ such that during voting, the votes for $A$ never dip below the votes for $B$.

I quite like the latter interpretation, because we really are counting two different things (votes for $A$ and $B$) and then expressing a relationship between them. It also allows us to directly prove that catalan(n) is equal to $1/(n+1) \binom{2n}{n}$ by reasoning about seqences of votes, called as ballot sequences

§ Ballot sequences

§ References

• Richard Stanley's slides