  We present a generating function and a closed counting formula in two
variables that enumerate a family of classes of permutations that avoid or
contain an increasing pattern of length three and have a prescribed number of
occurrences of another pattern of length three. This gives a refinement of some
previously studied statistics, most notably one by Noonan. The formula is also
shown to enumerate a family of classes of Dyck paths and Standard Young
Tableaux, and a bijection is given between the corresponding classes of these
two families of objects. Finally, the results obtained are used to solve an
optimization problem for a certain card game.
