  Chen et al. recently established bijections for $(d+1)$-noncrossing/
nonnesting matchings, oscillating tableaux of bounded height $d$, and
oscillating lattice walks in the $d$-dimensional Weyl chamber. Stanley asked
what is the total number of such tableaux of length $n$ and of any shape. We
find a determinant formula for the exponential generating function. The same
idea applies to prove Gessel's remarkable determinant formula for permutations
with bounded length of increasing subsequences. We also give short algebraic
derivations for some results of the reflection principle.
