Counting linear extensions of a partial order (linear orders compatible with the partial order) is a classical and computationally difficult problem in enumeration and computer science. A family of partial orders that are prevalent in enumerative and algebraic combinatorics come from Young diagrams of partitions and skew partitions. Their linear extensions are called standard Young tableaux. The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of partition shape. No such product formula exists for skew partitions. 

In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using ”excited diagrams” of  Ikeda-Naruse, Kreiman, Knutson-Miller-Yong in the context of equivariant cohomology. We prove Naruse’s formula algebraically and combinatorially in several different ways. Also, we show how excited diagrams give asymptotic results and product formulas for the enumeration of certain families of skew tableaux. Lastly, we give analogues of Naruse's formula in the context of equivariant K-theory.

This is joint work with Igor Pak and Greta Panova.