A Naples parking function with backward allowance 
, or 
-Naples parking function, is a sequence 
of preferences in 
for which all cars park under the following modification
of the parking function rule. If space 
is occupied, car 
checks spaces 
, 
, ..., down to 
in that order. It takes the first free space, or,
if these spaces are occupied, searches forward from 
(Christensen et al. 2020).
The case 
gives ordinary parking functions. For 
, all 
preference sequences succeed.
Unlike ordinary parking functions, success can
depend on the order of the preferences. For 
, 
succeeds, while 
fails.
Ferrari and Verciani (2026, Corollary 15) give a criterion for every rearrangement to succeed. Define
 |
For 
,
every rearrangement is a 
-Naples parking function iff each maximal
interval of consecutive integers in 
contains at most 
integers. For example, the preferences
give 
,
so they are not invariant under rearrangement for 
, but are for 
.
For 
,
Ferrari and Verciani (2026) call preferences complete when 
. Thus every threshold from 2 through 
has more preferences at or above it than spaces at or above
it.
