Vacillating Hecke tableaux and linked partitions

We introduce the structure of vacillating Hecke tableaux. By using the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis and Yong, we establish a one-to-one correspondence between vacillating Hecke tableaux and linked partitions, which arise in free probability theory. We define a Hecke diagram as a Young diagram possibly with a marked corner. A vacillating Hecke tableau is defined as a sequence of Hecke diagrams subject to a certain condition on addition and deletion of rook strips. The notion of a rook strip was introduced by Buch in the study of the Littlewood–Richardson rule for stable Grothendieck polynomials. We show that the crossing number and the nesting number of a linked partition can be determined by the maximal number of rows and the maximal number of columns of diagrams in the corresponding vacillating Hecke tableau. The proof relies on a theorem due to Thomas and Yong. As consequences, we confirm two conjectures on the distribution of the crossing number and the nesting number over linked partitions and ordinary partitions, respectively proposed by de Mier and Kim.