  Fill each box in a Young diagram with the number of paths from the bottom of
its column to the end of its row, using steps north and east. Then, any square
sub-matrix of this array starting on the south-east boundary has determinant
one. We provide a - to our knowledge - new bijective argument for this result.
Using the same ideas, we prove further identities involving these numbers which
correspond to an integral orthonormal basis of the inner product space with
Gram matrix given by the array in question. This provides an explicit answer to
a question (listed as unsolved) raised in Exercise 6.27 c) of Stanley's
Enumerative Combinatorics.
