A frustrated parking function of length 
is a sequence 
of preferred spots for 
cars arriving in order at 
numbered parking spaces. A car takes its preferred space when
it is free. Otherwise, it continues down the one-way street to the last, or largest-numbered,
unoccupied space, and the sequence is a frustrated parking
function if every car can park (Hallam et al. 2026).
Every frustrated parking function is a parking function, but the converse does not hold. Hallam et al. (2026) constructed a bijection
with height-labeled Dyck paths and proved that the number
of frustrated parking functions of length 
is
 |
Here 
denotes the double factorial. For 
, 1, ..., the values are 1, 1, 3, 15, 105, 945, 10395, ...
(OEIS A001147). They also proved that the number
of frustrated parking functions whose sets of lucky cars
or lucky spaces equal 
is 
, where 
is a Stirling
number of the second kind. The number with exactly 
lucky cars or lucky spaces is the corresponding entry of the
second-order Eulerian triangle,
whose rows begin 
, 
, 
, 
, ... (OEIS A008517).
The subclass in which every car or space after the first unlucky one is unlucky is
counted by the 
th Fubini number. For 
, 1, ..., these numbers are 1, 1, 3, 13, 75, 541, 4683, ...
(OEIS A000670).
