$k$th price auctions and Catalan numbers

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.