A386860 -id:A386860 - cp's OEIS frontend

A386861 The total number of big inversions in all parking functions of length n.

0, 0, 6, 150, 3240, 72030, 1720320, 44641044, 1260000000, 38584598580, 1277045637120, 45493302310170, 1737263457785856, 70841500488281250, 3073706745680363520, 141437374309896780360, 6881573428807007600640, 353052290206768309241544, 19051315200000000000000000

Offset: 1

Amanda Priestley, Aug 05 2025

A big inversion in a parking function (x_1,x_2,...,x_k) is a pair of integers i,j in [k] with i < j such that x_i - x_{j} >= 2.

			a(2) = 0 because in the 3 parking functions of length 2 (11, 12, 21), there are no inversions with difference strictly greater than one.
a(3) = 6 as in the 16 parking functions of length 3 (111, 112, 122, 121, 212, 221, 211, 123, 132, 213 312, 231, 321, 113, 131, 311)  312 has one big inversion, 231 has one, 321 has one, 131 has one, and 311 has 2. Thus, in the 16 parking functions of length 3 there are 6 total big inversions.

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.

a[n_]:=(n/4)*(n-1)*(n-2)*(n+1)^(n-2); Array[a,19] (* Stefano Spezia, Aug 06 2025 *)

a(n) = (n/4)*(n-1)*(n-2)*(n+1)^(n-2).

a(n) = A386860(n)*n/2. - 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

A386861 The total number of big inversions in all parking functions of length n.

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