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

Abstract

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.

CFT correlators, $$ \mathcal{W} $$-algebras and generalized Catalan numbers

“…for some function g(∆ L ). However, in the limit ∆ L → 0, we do find that F d=4 ∼ B d=4 (µ) log(1 − z) with B d=4 being the generating function of the number of linear extensions of the G[(1 k−1 ), (0 k−2 ), (0 k−2 )] posets (this is also the number of Young tableaux with restrictions; similar numbers were recently studied in [46]). 16 If we knew an algebraic equation satisfied by B d=4 , we could perhaps construct a differential equation whose solution would give the full minimal-twist stress tensor sector in d = 4 large-N CFTs in the limit ∆ L → 0.…”

CFT correlators, ${\cal W}$-algebras and Generalized Catalan Numbers

“…d=4 ∼ g(∆ L ) log k (1−z) for some function g(∆ L ). However, in the limit ∆ L → 0, we do find that F d=4 ∼ B d=4 (µ) log(1 − z) with B d=4 being the generating function of the number of linear extensions of the G[(1 k−1 ), (0 k−2 ), (0 k−2 )] posets (this is also the number of Young tableaux with restrictions; similar numbers were recently studied in [47]) 15 . If we knew an algebraic equation satisfied by B d=4 , we could perhaps construct a differential equation whose solution would give the full minimal-twist stress tensor sector in d = 4 large-N CFTs in the limit ∆ L → 0.…”