Enumeration of linear extensions (total orderings) of partially ordered sets (posets) is a classical topic in discrete mathematics. In the case of Young diagrams, these linear extensions are the Standard Young Tableaux, which give bases for the irreducible representations of the symmetric group, and are the favorite objects in algebraic combinatorics, with applications from statistical mechanics to algebraic geometry. 
In this talk we will review the classical results and background, and then generalize the problem to the enumeration of skew SYTs.  Recently Naruse discovered a positive compact formula for their enumeration from the study of equivariant Schubert calculus. We will prove a generalization of Naruse's Hook-Length formula for skew shapes via purely combinatorial methods. We will also show how such positive formula can be used for applications towards asymptotic enumeration and integrable models (lozenge tilings).
 
This is based on joint work with Alejandro Morales and Igor Pak.