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 110, 120, and 210.
It was shown that a_n also 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 100, 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.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
FORMULA
a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.0111684107126703379786799829348... - Vaclav Kotesovec, Oct 07 2021
EXAMPLE
The length 4 inversion sequences avoiding (110, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
The length 4 inversion sequences avoiding (100, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0121, 0122, 0123.
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1,
add(b(n-1, i-min(t, j)+2, abs(t-j)+1), j=1..i))
end:
a:= n-> b(n, 1$2):
seq(a(n), n=0..30); # Alois P. Heinz, Feb 21 2017
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[b[n - 1, i - Min[t, j] + 2, Abs[t-j]+1], {j, 1, i}]]; a[n_] := b[n, 1, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Megan A. Martinez, Feb 21 2017
EXTENSIONS
a(10)-a(25) from Alois P. Heinz, Feb 21 2017
STATUS
approved