  Schutzenberger's theorem for the ordinary RSK correspondence naturally
extends to Chen et. al's correspondence for matchings and partitions. Thus the
counting of bilaterally symmetric $k$-noncrossing partitions naturally arises
as an analogue for involutions. In obtaining the analogous result for
3-noncrossing partitions, we use a different technique to develop a Maple
package for 2-dimensional vacillating lattice walk enumeration problems. The
package also applies to the hesitating case. As applications, we find several
interesting relations for some special bilaterally symmetric partitions.
