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

Cyril Banderier (1) , Philippe Marchal (2) , Michael Wallner (3)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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.

Mots clés: binomial numbers; random generation; analytic combinatorics; Young tableau; density method; linear extensions of posets; generating function; D-finite function

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Probabilités [math.PR]

Cyril Banderier, Philippe Marchal, Michael Wallner. Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version). GASCOM 2018, Jun 2018, Athens, Greece. pp.60-68. ⟨hal-01795882⟩