  Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel
matrices of q-Catalan numbers are proven in two ways. One argument generalizes
the Bessenrodt-Stanley results for the Smith normal form of a certain
multivariate matrix that refines one studied by Berlekamp, Carlitz, Roselle,
and Scoville. The second argument, which uses orthogonal polynomials,
generalizes to a number of other Hankel matrices, Toeplitz matrices, and Gram
matrices. It gives new results for q-Catalan numbers, q-Motzkin numbers,
q-Schr\"oder numbers, q-Stirling numbers, q-matching numbers, q-factorials,
q-double factorials, as well as generating functions for permutations with
eight statistics.
