Abstract
A $K$-theoretic analogue of RSK insertion and the Knuth equivalence relations were introduced by Buch, Kresch, Shimozono, Tamvakis, and Yong (2006) and Buch and Samuel (2013), respectively. The resulting $K$-Knuth equivalence relations on words and increasing tableaux on $[n]$ has prompted investigation into the equivalence classes of tableaux arising from these relations. Of particular interest are the tableaux that are unique in their class, which we refer to as unique rectification targets (URTs). In this paper, we give several new families of URTs and a bound on the length of intermediate words connecting two $K$-Knuth equivalent words. In addition, we describe an algorithm to determine if two words are $K$-Knuth equivalent and to compute all $K$-Knuth equivalence classes of tableaux on $[n]$.