Extended boxed product and application to synchronized trees

We introduce a new technique to specify increasing labeled structures. For such objects, the boxed product as introduced by Greene is sufficient to efficiently specify the class, however for particular classes the size of the specification is very large. In particular, in the case of partially ordered sets, the calculus of the total orders compatibles with the poset (called linear extensions) can be tedious. We here developed an idea due to Stanley that uses a geometrical interpretation to calculate the linear extensions of a given poset. We will present a way to extend this idea to the symbolic method, and illustrate it with the example of specific increasing trees with exactly one repeated label, and show how to uniformly generate such structures.