A387047 Number of parking functions of size n with a big descent in the first position.

0, 0, 2, 25, 324, 4802, 81920, 1594323, 35000000, 857435524, 23219011584, 689292459245, 22272608433152, 778478027343750, 29273397577908224, 1178644785915806503, 50599804623580938240, 2307531308540969341448, 111411200000000000000000, 5677927131570439768106049

Offset: 1

Author

Amanda Priestley, Aug 14 2025

Keywords

• nonn

Comments

A big descent in a parking function (x_1,x_2,...,x_k) is a position i such that x_i - x_{i+1} >= 2.

Examples

			a(2)=0  because in the 3 parking functions of length 2 (11, 12, 21), there are 0 descents where the difference is strictly greater than one (and thus none in occur in the first position).
a(3)=2 because in the 16 parking functions of length 3, only 2 have a big descent occurring in the first position, 311 and 312.
a(4)=25 because in the 125 parking functions of length 4 there are 25 which have a big descent occurring in position 1. 3111, 4111, 3112, 3121, 4112, 4121, 4211, 3113, 3131, 3114, 3141, 4113, 4131, 3122, 4122, 4212, 4221, 3123, 3132, 3124, 3142, 4123, 4132, 4213, 4231.
		

Links

• Amanda Priestley, Table of n, a(n) for n = 1..100
• 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.

Crossrefs

Cf. A000272(n+1) (parking functions), A386860, A386015.

Programs

Mathematica

A387047[n_] := If[n < 2, 0, (n-2)*(n+1)^(n-2)/2];
Array[A387047, 25] (* Paolo Xausa, Aug 20 2025 *)

Formula

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

a(n) = A386860(n)/(n-1) for n >= 2.