OFFSET
0,3
COMMENTS
A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i < e_{i+1} <= e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 011 and 012.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..464
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.
FORMULA
a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n / n^alfa, where alfa = A073016 = Sum_{k>=1} 1/binomial(2*k, k) = 1/3 + 2*Pi/3^(5/2) = 0.73639985871871507790... and c = 2.21611825460684222558745179... - Vaclav Kotesovec, Oct 19 2019
EXAMPLE
The a(4)=11 length 4 inversion sequences avoiding the consecutive patterns 011 and 012 are 0000, 0100, 0010, 0020, 0001, 0101, 0021, 0002, 0102, 0003, and 0103.
MAPLE
# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
`if`(t and i < x, 0, b(n - 1, i, i <= x)), i = 0 .. n - 1))
end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);
MATHEMATICA
b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i < x, 0, b[n - 1, i, i <= x]], {i, 0, n - 1}]];
a[n_] := b[n, -1, False];
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan S. Auli, Oct 16 2019
STATUS
approved