A386015 Number of parking functions of size n with a descent in the first position.

0, 1, 6, 50, 540, 7203, 114688, 2125764, 45000000, 1071794405, 28378791936, 827150951094, 26322173602816, 908224365234375, 33776997205278720, 1347022612475207432, 57346445240058396672, 2595972722108590509129, 124518400000000000000000, 6308807923967155297895610, 336682260736692839281065984

Offset

1,3

Links

Paolo Xausa, Table of n, a(n) for n = 1..350

Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, Inversions in parking functions, arXiv:2508.11587 [math.CO], 2025.

Wikipedia, Parking function.

Formula

a(n) = (n/2)*(n+1)^(n-2) for n >= 2.

a(n) = A085389(n) / 2 for n >= 2.

Example

For n=1 the only parking function is 1 and it does not have a descent in the first position so a(1)=0.
For n=2 there are 3 parking functions: 11,12,21. Among them only 21 has a descent in the first position so a(2)=1.
For n=3 there are 16 parking functions: 111,112,121,211,122,212,221,113,131,311,123,132,213,231,312,321. Of these, 211,212,311,213,312, and 321 have a descent in the first position so a(3)=6.

Mathematica

A386015[n_] := If[n == 1, 0, n*(n+1)^(n-2)/2];
Array[A386015, 25] (* Paolo Xausa, Aug 07 2025 *)

Crossrefs

Cf. A000272, A085389.

Sequence in context: A300989 A105617 A297926 * A094072 A361193 A363315

Adjacent sequences: A386012 A386013 A386014 * A386016 A386017 A386018

Keyword

nonn,easy

Author

Gabe Udell, Jul 14 2025

Status

approved