kth price auctions and Catalan numbers

This paper shows that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction with k≥3 can be represented as a finite series of k−2 terms whose ℓth term involves the ℓth 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 kth price auction for a non-uniform distribution.