 These order-preserving maps were first introduced by Richard P. Stanley while studying ordered structures and partitions as a Ph.D. student at Harvard University in 1971 under the guidance of Gian-Carlo Rota.Let formula_2 be a finite poset with formula_3 elements denoted formula_4, and let formula_5 be a chain formula_1 elements.A map formula_7 is order-preserving if formula_8 implies formula_9.