  This paper establishes an interesting link between $k$th price auctions and
Catalan numbers by showing that for distributions that have linear density, the
bid function at any symmetric, increasing equilibrium of a $k$th price auction
with $k\geq 3$ can be represented as a finite series of $k-2$ terms whose
$\ell$th term involves the $\ell$th Catalan number. Using an integral
representation of Catalan numbers, together with some classical combinatorial
identities, we derive the closed form of the unique symmetric, increasing
equilibrium of a $k$th price auction for a non-uniform distribution.
