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

0,3

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<j<k) such that e_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.

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.

Formula

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

Example

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.

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 *)

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.

Sequence in context: A150245 A150246 A055700 * A150247 A150248 A105871

Adjacent sequences: A279560 A279561 A279562 * A279564 A279565 A279566

Keyword

nonn

Author

Megan A. Martinez, Feb 09 2017

Extensions

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

Status

approved