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Dyck paths are one of the most important objects in enumerativecombinatorics, and there are many papers devoted to counting selected familiesof Dyck paths. Here we present two approaches for the automatic counting ofmany such families, using both a "dumb" approach (driven by numeric dynamicprogramming) that often works in practice, and a "clever" approach, needed forlarger problems, driven by "symbolic" dynamic programming. Both approaches arefully automated and implemented in Maple.

Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming