Lattices with Series-Parallel and Interval Order and a Generalization of Catalan Numbers

We obtain a formula for the number of finite lattices of a given height and cardinality that have a series-parallel and interval order. Our approach is to consider a naturally defined class of \(m\) nested intervals on an \(m+k\)-element chain, and we show that there are \(\binom{m-1}{k-1}\gamma(m+1)\) such sets of nested intervals. Here, \(\gamma(m+1)\) denotes the Catalan number \(\frac{1}{m+1}\binom{2m}{m}\).