  We call a finite dimensional algebra A S-connected if the projective
dimensions of the simple A-modules form an interval. We prove that a Nakayama
algebra A is S-connected if and only if A is quasi-hereditary. We apply this
result to improve an inequality for the global dimension of quasi-hereditary
Nakayama algebras due to Brown. We furthermore classify the Nakayama algebras
where equality is attained in Brown's inequality and show that they are
enumerated by the even indexed Fibonacci numbers if the algebra is cyclic and
by the odd indexed Fibonacci numbers if the algebra is linear.
