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

Abstract

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 $(\genfrac{}{}{0ex}{}{m-1}{k-1})\gamma (m+1)$ such sets of nested intervals. Here, $\gamma (m+1)$ denotes the Catalan number $\frac{1}{m+1}(\genfrac{}{}{0ex}{}{2m}{m})$.