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

1,3

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.

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.

Formula

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

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A085830 A270346 A367506 * A212022 A198710 A074209

Adjacent sequences: A387044 A387045 A387046 * A387048 A387049 A387050

Keyword

nonn

Author

Amanda Priestley, Aug 14 2025

Status

approved