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

Author

Gabe Udell, Jul 14 2025

Keywords

• nonn
• easy

Examples

			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.
		

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.

Crossrefs

Cf. A000272, A085389.

Programs

Mathematica

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

Formula

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

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