Semiorders and Riordan Numbers

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 comes from the fact that each element x ∈ X can be identified with an interval on the real line. All intervals are the same length, and two intervals intersect if and only if their corresponding elements are incomparable. If the intervals for a and b do not intersect, and the interval for a lies to the left of the interval for b, then a ≺ b. We may assume without loss of