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) used height-labeled Dyck paths to prove 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). A car is called lucky if it
parks in its preferred space, and a space is lucky if it is occupied by a lucky car
(Hallam et al. 2026). They also proved that the number of frustrated parking
functions whose set of lucky cars equals 
is 
, and the same count holds for those whose set
of lucky spaces equals 
, 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 subclasses in which every car after the first unlucky car is unlucky, or every
space after the first unlucky space is unlucky, are each counted by the 
th ordered Bell number
(also called a Fubini number). For 
, 1, ..., these numbers are 1, 1, 3, 13, 75, 541, 4683, ...
(OEIS A000670).
