On the Number of Fixed-Length Semiorders

A semiorder is a partially ordered set $P$ with two certain forbidden induced subposets. This paper establishes a bijection between $n$-element semiorders of length $H$ and $(n+1)$-node ordered trees of height $H+1$. This bijection preserves not only the number of elements, but also much additional structure. Based on this correspondence, we calculate the generating functions and explicit formulas for the numbers of labeled and unlabeled $n$-element semiorders of length $H$. We also prove several concise recurrence relations and provide combinatorial proofs for special cases of the explicit formulas.