Schröder partitions and Schröder tableaux

We introduce the notions of Schröder shape and of Schröder tableau, which provide some kind of analogs of the classical notions of Young shape and Young tableau. We investigate some properties of the partial order given by containment of Schröder shapes. Then we propose an algorithm which is the natural analog of the well known RS correspondence for Young tableaux, and we characterize those permutations whose insertion tableaux have some special shapes. We end our paper with a few suggestions for possible further work.