  Nice formulae for plane partitions with bounded size of parts (or boxed plane
partitions), which generalize the norm-trace generating function by Stanley and
the trace generating function by Gansner, are exhibited. The derivation of the
nice formulae is based on lattice path combinatorics of biorthogonal
polynomials, especially of the little $q$-Laguerre polynomials and a
generalization of the little $q$-Laguerre polynomials. A summation formula
which generalizes the $q$-Chu-Vandermonde identity is also shown and utilized
to prove the orthogonality of the generalized little $q$-Laguerre polynomials.
