A386015 -id:A386015 - cp's OEIS frontend

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

0, 0, 4, 75, 1296, 24010, 491520, 11160261, 280000000, 7716919716, 232190115840, 7582217051695, 267271301197824, 10120214355468750, 409827566090715136, 17679671788737097545, 809596873977295011840, 39228032245196478804616, 2005401600000000000000000, 107880615499838355594014931

Offset: 1

Amanda Priestley, Aug 05 2025

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.

			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.
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.

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.

Cf. A000272(n+1) (parking functions), A333829, A386015, A386861.

A386860[n_] := Binomial[n-1, 2]*(n+1)^(n-2);
Array[A386860, 20] (* Paolo Xausa, Aug 07 2025 *)

a(n) = binomial(n-1,2)*(n+1)^(n-2).

a(n) = A386861(n)*2/n. - Paolo Xausa, Aug 07 2025

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

Original entry on oeis.org

0, 0, 2, 25, 324, 4802, 81920, 1594323, 35000000, 857435524, 23219011584, 689292459245, 22272608433152, 778478027343750, 29273397577908224, 1178644785915806503, 50599804623580938240, 2307531308540969341448, 111411200000000000000000, 5677927131570439768106049

Offset: 1

Amanda Priestley, Aug 14 2025

nonn

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.

			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.

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.

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

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

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

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

cp's OEIS Frontend

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

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