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

%I A387047 #27 Aug 20 2025 10:57:03
%S A387047 0,0,2,25,324,4802,81920,1594323,35000000,857435524,23219011584,
%T A387047 689292459245,22272608433152,778478027343750,29273397577908224,
%U A387047 1178644785915806503,50599804623580938240,2307531308540969341448,111411200000000000000000,5677927131570439768106049
%N A387047 Number of parking functions of size n with a big descent in the first position.
%C A387047 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 A387047 Amanda Priestley, <a href="/A387047/b387047.txt">Table of n, a(n) for n = 1..100</a>
%H A387047 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 A387047 a(n) = (n-2)/2*(n+1)^(n-2) for n >= 2.
%F A387047 a(n) = A386860(n)/(n-1) for n >= 2.
%e A387047 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 A387047 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 A387047 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 A387047[n_] := If[n < 2, 0, (n-2)*(n+1)^(n-2)/2];
%t A387047 Array[A387047, 25] (* _Paolo Xausa_, Aug 20 2025 *)
%Y A387047 Cf. A000272(n+1) (parking functions), A386860, A386015.
%K A387047 nonn
%O A387047 1,3
%A A387047 _Amanda Priestley_, Aug 14 2025