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}. Alternatively, we can describe this as the set of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..465
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^(2*Pi/3^(3/2)), where c = 0.75844492121718325018323312623016463... - Vaclav Kotesovec, Oct 17 2019
EXAMPLE
The a(4)=4 length 4 inversion sequences avoiding the consecutive patterns 000, 001, 011, 012 are 0100, 0101, 0102, 0103.
The a(5)=6 length 5 inversion sequences are 01010, 01020, 01021, 01030, 01031, 01032.
MAPLE
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=1..n))
end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..24); # Alois P. Heinz, Oct 14 2019
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, 1, n}]];
a[n_] := b[n, 0, False];
a /@ Range[0, 24] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan S. Auli, Oct 13 2019
EXTENSIONS
Terms a(11)..a(16) from Joerg Arndt, Oct 14 2019
a(17)-a(24) from Alois P. Heinz, Oct 14 2019
STATUS
approved