  We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two
row arrays of multisets. This generalization leads to new enumerative results
that have representation theoretic interpretations as decompositions of
centralizer algebras and the spaces they act on. In addition, restrictions on
the multisets lead to further identities and representation theory analogues.
For instance, we obtain a bijection between words of length $k$ with entries in
$[n]$ and pairs of tableaux of the same shape with one being a standard Young
tableau of size $n$ and the other being a standard multiset tableau of content
$[k]$. We also obtain an algorithm from partition diagrams to pairs of a
standard tableau and a standard multiset tableau of the same shape, which has
the remarkable property that it is well-behaved with respect to restricting a
representation to a subalgebra. This insertion algorithm matches recent
representation-theoretic results of Halverson and Jacobson.
