  If A is an artin algebra, G\'elinas has introduced an interesting upper bound
for the finitistic dimension of A, namely the delooping level del A. We assert
that for any Nakayama algebra, its finitistic dimension is equal to del A. This
yields also a new proof that the finitistic dimension of A and of its opposite
algebra are equal, as shown recently by Sen. If S is a simple module, let e(S)
be the minimum of the projective dimension of S and of its injective envelope
(one of these numbers has to be finite); and e*(S) the minimum of the injective
dimension of S and of its projective cover. Then the finitistic dimension of A
is the maximum of the numbers e(S) as well as the maximum of the numbers
e^*(S).
  Using suitable syzygy modules, we construct a permutation h of the simple
modules S such that e*(h(S)) = e(S). In particular, this shows for any natural
number z, that the number of simple modules S with e(S) = z is equal to the
number of simple modules S' with e(S') = z.
