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 100, 110, 120, and 210.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
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) ~ c * (1 + sqrt(2))^(2*n) / n^(3/2), where c = 0.066085708825649431003670013119332303648755519420440375... - Vaclav Kotesovec, Oct 07 2021
EXAMPLE
The length 4 inversion sequences avoiding (100, 110, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
MAPLE
b:= proc(n, i, m) option remember; `if`(n=0, 1, add((h->
b(n-1, i-h+1, max(m, j)-h))(max(0, min(m-1, j))), j=1..i))
end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..30); # Alois P. Heinz, Feb 23 2017
MATHEMATICA
b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[b[n-1, i-#+1, Max[m, j]-#]& @ Max[0, Min[m-1, j]], {j, 1, i}]]; a[n_] := b[n, 1, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Megan A. Martinez, Feb 09 2017
EXTENSIONS
a(10)-a(25) from Alois P. Heinz, Feb 23 2017
STATUS
approved