A Family of 0-Simple Semihypergroups Related to Sequence A000070

For any integer n ≥ 2, let R0(n + 1) be the class of 0-semihypergroups H of size n + 1 such that {y} ⊆ xy ⊆ {0, y} for all x, y ∈ H − {0}, all subsemihypergroups K ⊆ H are 0-simple and, when |K | ≥ 3, the fundamental relation βK is not transitive. We determine a transversal of isomorphism classes of semihypergroups in R0(n + 1) and we prove that its cardinality is the (n + 1)-th term of sequence A000070 in [21], namely, ∑n k=0 p(k), where p(k) denotes the number of non-increasing partitions of integer k.