A386861 The total number of big inversions in all parking functions of length n.

0, 0, 6, 150, 3240, 72030, 1720320, 44641044, 1260000000, 38584598580, 1277045637120, 45493302310170, 1737263457785856, 70841500488281250, 3073706745680363520, 141437374309896780360, 6881573428807007600640, 353052290206768309241544, 19051315200000000000000000

Offset: 1

Author

Amanda Priestley, Aug 05 2025

Keywords

• nonn
• easy

Comments

A big inversion in a parking function (x_1,x_2,...,x_k) is a pair of integers i,j in [k] with i < j such that x_i - x_{j} >= 2.

Examples

			a(2) = 0 because in the 3 parking functions of length 2 (11, 12, 21), there are no inversions with difference strictly greater than one.
a(3) = 6 as in the 16 parking functions of length 3 (111, 112, 122, 121, 212, 221, 211, 123, 132, 213 312, 231, 321, 113, 131, 311)  312 has one big inversion, 231 has one, 321 has one, 131 has one, and 311 has 2. Thus, in the 16 parking functions of length 3 there are 6 total big inversions.
		

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.

Programs

Mathematica

a[n_]:=(n/4)*(n-1)*(n-2)*(n+1)^(n-2); Array[a,19] (* Stefano Spezia, Aug 06 2025 *)

Formula

a(n) = (n/4)*(n-1)*(n-2)*(n+1)^(n-2).

a(n) = A386860(n)*n/2. - Paolo Xausa, Aug 07 2025