A279563 Number of length n inversion sequences avoiding the patterns 102, 201, and 210.

1, 1, 2, 6, 22, 85, 328, 1253, 4754, 17994, 68158, 258808, 985906, 3768466, 14451386, 55585014, 214377618, 828795169, 3211030684, 12464308997, 48465092366, 188733879657, 735977084412, 2873525548315, 11231884145434, 43947466923095, 172115939825516

Offset: 0

Author

Megan A. Martinez, Feb 09 2017

Keywords

• nonn

Comments

A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_j <> e_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 102, 201, and 210.

Examples

			The length 4 inversion sequences avoiding (102, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1664
• Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Crossrefs

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, n!,
      ((2*(12*n^3-91*n^2+213*n-149))*a(n-1)
      -(3*(21*n^3-162*n^2+392*n-291))*a(n-2)
      +(2*(33*n^3-257*n^2+633*n-484))*a(n-3)
      -(4*(2*n-7))*(3*n^2-13*n+13)*a(n-4))
       / ((n-1)*(3*n^2-19*n+29)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 22 2017

Mathematica

a[n_] := a[n] = If[n < 4, n!, ((2*(12*n^3 - 91*n^2 + 213*n - 149))*a[n-1] - (3*(21*n^3 - 162*n^2 + 392*n - 291))*a[n-2] + (2*(33*n^3 - 257*n^2 + 633*n - 484))*a[n-3] - (4*(2*n - 7))*(3*n^2 - 13*n + 13)*a[n-4]) / ((n - 1)*(3*n^2 - 19*n + 29))]; Array[a, 30, 0] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

Formula

a(n) ~ 4^n / (3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2021

G.f.: (2 - 15*x + 32*x^2 - 16*x^3 + x * (1 - 2*x) * (1 + 2*x) * (1 - 4*x)^(1/2)) / (2 * (1 - x)^2 * (1 - 2*x) * (1 - 4*x)). - Nathan J. Britt, Jun 08 2025

Extensions

a(10)-a(26) from Alois P. Heinz, Feb 22 2017