The monoid of ordered partitions of a natural number

The sequences of non-negative integers form a monoid, a natural submonoid of which has elements corresponding to order-preserving transformations of a finite chain. This, in turn, has a submonoid whose elements are ordered partitions of a natural number. A presentation for the last monoid is given, and the inclusion poset of principal right ideals is described. The poset of principal left ideals has a recursive structure that gives rise to an interesting sequence of numbers.