5,1

This sequence enumerates parking functions with lucky fifth spot (where a lucky spot is one which is parked in by a car which prefers that spot).

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.

a(n) = (3/5)*(n+1)^(n-1) - (1/30)*(118*n^3 - 531*n^2 + 659*n - 192)*(n-4)^(n-4).

For clarity, we write parentheses around parking functions. For n = 6, there are a(6) = 8733 solutions. An example of a parking function of order 6 with a lucky fifth spot is (1,4,1,5,2,2); here, the fourth car parks in the fifth spot which is its preferred spot. This parking function contributes to our count. A non-example is the parking function (1,1,1,1,1,5); here, the fifth car parks in the fifth spot, but its preference is spot 1. This parking function does not contribute to our count.

a[n_]:=(3/5)*(n+1)^(n-1)-(1/30)*(118*n^3-531*n^2+659*n-192)*(n-4)^(n-4); Array[a, 17, 5] (* Stefano Spezia, Jun 26 2024 *)

(Python)

def A372845(n): return (18*(n+1)**(n-1)-(n*(n*(59*((n<<1)-9))+659)-192)*(n-4)**(n-4))//30 # Chai Wah Wu, Jun 26 2024

Cf. A000272 (for first spot), A372842 (for second spot), A372843 (for third spot), and A372844 (for fourth spot).

Sequence in context: A143026 A238700 A064781 * A250832 A250827 A055868

Adjacent sequences: A372842 A372843 A372844 * A372846 A372847 A372848

Kimberly P. Hadaway, Jun 26 2024

approved