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

Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version)