Combinatorial results for semigroups of order-preserving partial transformations.
This paper gathers together some combinatorial results for the semigroup \(S=\mathcal{PO}_n\), the semigroup of all partial order-preserving mappings on a finite \(n\)-chain. Counting the numbers of mappings with domains and ranges of particular cardinalities eventually leads to a second-order nonlinear recurrence for \(|S|\). A similar set of results is obtained for \(e_n\), the number of idempotents. On this occasion the solution is explicitly determined in terms of powers of the golden ratio and curiously it transpires that \(e_n-1\) is a multiple of five. The final section proves counting results on the sizes of classes determined by Green's relations on \(S\).
scientific article; zbMATH DE number 2102503
scientific article