  In this article, we consider a generalization of Young tableaux in which we
allow some consecutive pairs of cells with decreasing labels. We show that this
leads to a rich variety of combinatorial formulas, which suggest that these new
objects could be related to deeper structures, similarly to the ubiquitous
Young tableaux. Our methods rely on variants of hook-length type formulas, and
also on a new efficient generic method (which we call the density method) which
allows not only to generate constrained combinatorial objects, but also to
enumerate them. We also investigate some repercussions of this method on the
D-finiteness of the generating functions of combinatorial objects encoded by
linear extension diagrams, and give a limit law result for the average number
of local decreases.
