Enumeration of bilaterally symmetric 3-noncrossing partitions

Schützenberger’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.