  We study pfaffian analogues of immanants, which we call pfaffinants. Our main
object is the TL-pfaffinants which are analogues of Rhoades and Skandera's
TL-immanants. We show that TL-pfaffinants are positive when applied to planar
networks and explain how to decompose products of complementary pfaffians in
terms of TL-pfaffinants. We conjecture in addition that TL-pfaffinants have
positivity properties related to Schur Q-functions.
