A387047 - OEIS

%I #27 Aug 20 2025 10:57:03

%S 0,0,2,25,324,4802,81920,1594323,35000000,857435524,23219011584,

%T 689292459245,22272608433152,778478027343750,29273397577908224,

%U 1178644785915806503,50599804623580938240,2307531308540969341448,111411200000000000000000,5677927131570439768106049

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

%C 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.

%H Amanda Priestley, <a href="/A387047/b387047.txt">Table of n, a(n) for n = 1..100</a>

%H Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Amanda Priestley, and Gabe Udell, <a href="https://arxiv.org/abs/2508.11587">Inversions in parking functions</a>, arXiv:2508.11587 [math.CO], 2025.

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

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

%e 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).

%e 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.

%e 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.

%t A387047[n_] := If[n < 2, 0, (n-2)*(n+1)^(n-2)/2];

%t Array[A387047, 25] (* _Paolo Xausa_, Aug 20 2025 *)

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

%K nonn

%O 1,3

%A _Amanda Priestley_, Aug 14 2025