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_j >= e_k and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 000, 010, 100, 110, 120, and 210.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..600
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
FORMULA
a(n) ~ c * 4^n / n^(3/2), where c = 0.0549097036253448014962069269284638611865763295943683310517... - Vaclav Kotesovec, Oct 07 2021
EXAMPLE
For n=3, the inversion sequences are 001, 002, 011, 012.
MAPLE
b:= proc(n, i, m) option remember; `if`(i=0, 1, add(
b(n-min(m, j), i-1, abs(m-j)), j=1..n-i+1))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30); # Alois P. Heinz, Dec 15 2016
MATHEMATICA
b[n_, i_, m_] := b[n, i, m] = If[i == 0, 1, Sum[b[n - Min[m, j], i - 1, Abs[m - j]], {j, 1, n - i + 1}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Megan A. Martinez, Dec 14 2016
EXTENSIONS
a(10)-a(28) from Alois P. Heinz, Dec 14 2016
Name and description corrected by Nicholas R. Beaton, May 02 2024
STATUS
approved