  We give an upper bound for the index of certain Lie algebras, called of
seaweed type, introduced by V. Dergachev, A. Kirillov and D. Panyushev. We
deduce from this a conjecture of D. Panyushev stated in "Inductive formulas for
the index of seaweed Lie algebra, Moscow Math. Journal (2001) 221-241". We
conjecture that our upper bound gives in fact a formula for the index of such
Lie algebras, and we prove it in certain cases. Finally, we obtain a result
concerning the index of parabolic subalgebras of a semisimple Lie algebra.
