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A386860 The total number of big descents in all parking functions of length n.

%I A386860 #35 Aug 19 2025 16:36:43
%S A386860 0,0,4,75,1296,24010,491520,11160261,280000000,7716919716,
%T A386860 232190115840,7582217051695,267271301197824,10120214355468750,
%U A386860 409827566090715136,17679671788737097545,809596873977295011840,39228032245196478804616,2005401600000000000000000,107880615499838355594014931
%N A386860 The total number of big descents in all parking functions of length n.
%C A386860 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 A386860 Amanda Priestley, <a href="/A386860/b386860.txt">Table of n, a(n) for n = 1..100</a>
%H A386860 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 A386860 a(n) = binomial(n-1,2)*(n+1)^(n-2).
%F A386860 a(n) = A386861(n)*2/n. - _Paolo Xausa_, Aug 07 2025
%e A386860 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.
%e A386860 a(3) = 4 as of the 16 parking functions of length 3 (111, 112, 122, 121, 212, 221, 211, 123, 132, 213, 312, 231, 321, 113, 131, 311) the parking functions (131, 311, 312, 231) all each have one big descent. Thus the total number of big descents in all parking functions of length 3 is 4.
%t A386860 A386860[n_] := Binomial[n-1, 2]*(n+1)^(n-2);
%t A386860 Array[A386860, 20] (* _Paolo Xausa_, Aug 07 2025 *)
%Y A386860 Cf. A000272(n+1) (parking functions), A333829, A386015, A386861.
%K A386860 nonn,easy
%O A386860 1,3
%A A386860 _Amanda Priestley_, Aug 05 2025