Combinatorial Identities for Vacillating Tableaux

Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions. In this paper, we further investigate the enumeration of vacillating tableaux and derive multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux, simplified vacillating tableaux, and limiting vacillating tableaux.