Journal of Integer Sequences, Vol. 10 (2007), Article 07.7.6

In this paper, we define a class of semiorders (or unit interval orders) that arose in the context of polyhedral combinatorics. In the first section of the paper, we will present a pure counting argument equating the number of these interesting (connected and irredundant) semiorders on n + 1 elements with the nth Riordan number. In the second section, we will make explicit the relationship between the interesting semiorders and a special class of Motzkin paths, namely, those Motzkin paths without horizontal steps of height 0, which are known to be counted by the Riordan numbers. 1 Counting Interesting Semiorders We begin with some basic definitions. Definition 1. A partially ordered set (X, ≺) is a semiorder if it satisfies the following two properties for any a,b,c,d ∈ X. • If a ≺ b and c ≺ d, a ≺ d or c ≺ b. • If a ≺ b ≺ c, then d ≺ c or a ≺ d. Semiorders are also known as unit interval orders in the literature. This name come