  Quantum determinants and Pfaffians or permanents and Hafnians are introduced
on the two parameter quantum general linear group. Fundamental identities among
quantum Pf, Hf, and det are proved in the general setting. We show that there
are two special quantum algebras among the quantum groups, where the quantum
Pfaffians have integral Laurent polynomials as coefficients. As a consequence,
the quantum Hafnian is computed by a closely related quantum permanent and
identical to the quantum Pfaffian on this special quantum algebra.
