  In this note, we introduce a statistic on Motzkin paths that describes the
rank generating function of Bruhat order for involutions. Our proof relies on a
bijection introduced by Philippe Biane from permutations to certain labeled
Motzkin paths and a recently introduced interpretation of this rank generating
function in terms of visible inversions. By restricting our identity to
fixed-point-free (FPF) involutions, we recover an identity due to Louis
Billera, Lionel Levine and Karola M\'esz\'aros with a previous bijective proof
by Matthew Watson. Our work sheds new light on the Ethiopian dinner game.
