The number of partial orders of fixed width

Abstract

We investigate the approximate number of n-element partial orders of width k, for each fixed k. We show that the number of width 2 partial orders with vertex set {1, 2, ..., n} is

$$\frac{{(2n + 1)!}}{{(n + 1)!}}{\text{ }}\left( {\frac{4}{{25}} + o(1)} \right) = n!\frac{{4^n }}{{n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }}{\text{ }}\frac{8}{{25\sqrt \pi }}{\text{ }}(1 + o(1))$$

as n→∞. We obtain a similar result for the number of unlabelled n-vertex width 2 partial orders. For each fixed k≥2, we show that the number of width k partial orders with vertex set {1, ... n} is within a polynomial (in n) factor of n!4^n(k-1).