A374533 a(n) is the number of parking functions of order n where the (n-1)-st spot is lucky.

3, 11, 74, 708, 8733, 131632, 2342820, 48068672, 1116809255

Offset

2,1

Comments

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).

Links

Table of n, a(n) for n=2..10.

Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 10.

Example

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.

Crossrefs

Second diagonal of A374756.

Sequence in context: A350834 A054461 A342433 * A203772 A290025 A239658

Adjacent sequences: A374530 A374531 A374532 * A374534 A374535 A374536

Keyword

nonn,more,changed

Author

Kimberly P. Hadaway, Jul 10 2024

Status

approved