%I #13 Dec 27 2024 08:55:29
%S 3,11,74,708,8733,131632,2342820,48068672,1116809255
%N a(n) is the number of parking functions of order n where the (n-1)-st spot is lucky.
%C This sequence enumerates parking functions with lucky penultimate spot (where a lucky spot is one which is parked in by a car which prefers that spot).
%H Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See p. 10.
%e For clarity, we write parentheses around parking functions. For n = 3, the a(3) = 11 solutions are the parking functions of length 3 with a lucky second spot: (1,2,1),(1,2,2),(1,2,3),(1,3,2),(2,1,1),(2,1,2),(2,1,3),(2,2,1),(2,3,1),(3,1,2),(3,2,1). There are 5 parking functions of length 3 which do not have a lucky second spot: (1,1,1),(1,1,2),(1,1,3),(1,3,1),(3,1,1). For all of these, the car which parks in the second spot did not prefer the second spot; these parking functions do not contribute to our count.
%Y Second diagonal of A374756.
%K nonn,more
%O 2,1
%A _Kimberly P. Hadaway_, Jul 10 2024