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 100, 101, and 201.
LINKS
Nicholas R. Beaton, Table of n, a(n) for n = 0..1000 (terms 0..32 from Vaclav Kotesovec)
Nathan Britt and Nicholas Beaton, Completing the enumeration of inversion sequences avoiding triples of relations, arXiv:2512.21943 [math.CO], 2025. See p. 3, Table 1 and p. 22.
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 * n^g * 9^n, where g = -5.401688... and C = 88.712... - Vaclav Kotesovec, Jan 13 2026 (after Nicholas R. Beaton, constants from the article refined)
EXAMPLE
The length 4 inversion sequences avoiding (100,101,201) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0102, 0103, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
MAPLE
b:= proc(n, i, s, m) option remember;
`if`(n=0, 1, add(b(n-1, i+1, s minus {$j..m-
`if`(j=m, 1, 0)} union {i+1}, max(m, j)), j=s))
end:
a:= n-> b(n, 1, {1}, 0):
seq(a(n), n=0..15); # Alois P. Heinz, Feb 22 2017
MATHEMATICA
b[n_, i_, s_, m_] := b[n, i, s, m] = If[n == 0, 1, Sum[b[n-1, i+1, s ~Complement~ Range[j, m - If[j == m, 1, 0]] ~Union~ {i+1}, Max[m, j]], {j, s}]];
a[n_] := b[n, 1, {1}, 0];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 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 22 2017
a(26)-a(29) from Vaclav Kotesovec, Oct 07 2021
STATUS
approved